BULLETIN

UNITED STATES

GEOLOGICAL SURVEY

2STo. 48

WASHINGTON

GOVERNMENT PRINTING OFFICE 

1SSS

UNITED STATES GEOLOGICAL SURVEY

J. W. POWELL, DIRECTOR

ON THE FORM AND POSITION

OH'

THE SEA LEVEL

WITH SPECIAL UEKEIIENCE TO ITS DEPENDENCE OX SUPEKiriCIAL

MASSES SYMMETRICALLY DISPOSED ABOUT A NOKMAL

TO THE EAKTH'S SUKFACE

ROBERT SIMPSON WOODWARD

WASHINGTON . -

GOVERNMENT PRINTING OFFICE

1888

CONTENTS.

Page. 

Key to mathematical symbols ............................................... 9

Letter of transmittal........................................................ 13

I. Introduction..... .................................................... 15

1. Form and dimensions of sea-level surface of earth. Close approximation of oblate spheroid. Eolation of actual sea surface or 

geoid to spheroidal surface. A knowledge required of form of 

geoid by geodesy, of variations in form and position by geology. 

Difficulties in way of improved theory ......................... 15

2. Class of problems discussed in this paper........................... 16

3. R6sum6 of results attained......................................... 17

A. THEORY.

II. Mathematical statement of problem ............ ...................... 18

4. Fundamental principle and equation............................... 18

5. Dimensions of earth's ellipsoid and sphere of equal volume.......... 19

6. Derivation of equation of disturbed surface ........................ 19

III. Evaluation of potential of disturbing mass of uniform thickness....... 21

7. Determination of potential iu terms of rectangular and polar co-ordinates ......................................................... 21

8. Transformation and reduction to single integration of elliptic forms. 22

9. Discussion and further transformation ............................. 24

10. Special values of the integrals and'corresponding values of the potential ........................................................ 25

(a) For a point of the disturbed surface at the center of the disturbing mass.................................................... 25

(Z») For a point of the disturbed surface at the border of the disturbing mass.................................................... 25

(c) For a point of the disturbed surface 180° from the center of the

disturbing mass............................................. 26

(d) Potential of a spherical shell .................................. 26

IV. Degree of approximation of the expressions for the potential of the disturbing mass................ .................................

11. Exact expression for potential of complete spherical shell.... .:....

12. Degree of approximation of expression for potential at center of disturbing mass.... .............. ...... ............ .... .......... 27

13. Degree of approximation of expression for potential at border of disturbing maSs.................................................. 28

14. Degree of approximation of expression for potential at point 180°

from center of disturbing mass................................. 30

V. Development of potential of disturbing mass in series of spherical harmonics...... .................................................. 30

15. Remffrks on expressions for potential previously derived and on

those to bo considered......................................... 30

16. Expansion of potential function in series and integration of separate terms .................................................... 31

(89) ' 5
0 CONTENTS.

Page.

17. Discussion and derivation of approximate forms. Harmonic development of elliptic integrals Ii and J2 1....................... 34

VI. Effect of re-arranged free water....................................... 35

18. Eemark on difficulty of obtaining exact expression for effect of rearranged free water. Derivation of expression for an effect which 

will exceed probable actual eifcct.............................. 35

VII. Evaluation of constants Fo and U0 in equation to disturbed surface.... 37

19. Statement of principle involved in determination of constants Fa

and Uo and their evaluation....... ............................ 37

(a) Values of F0 and Uo found by means of property of spherical

harmonics .................................................. 38

(&) Value of Fo found by direct integration .......................j 33

VIII. Equations of disturbed surface........................................ 40

20. Equations of disturbed surface when effect of re-arranged water is

neglected and when that effect is considered ................... 40

21. Discussion of equations........................................... 41

22. Special values of the elevation of the disturbed surface at the center, at the border, and 180° from the center of the disturbing 

mass ............. .............................\.............. 41

23. Angular radial extent of masses of uniform thickness requisite to

"produce maximum elevation of disturbed surface .............. 42

24. Effect of re- arranged free water.................................. 42

IX. Evaluation of the definite integrals 7i and 72 ......................... 43

25. Expansion of Ji in series ...................... .................. 43

26. Expansion of J2 in series ........................................ 45

27. Additional expansion of J2 for case when attracted point is near

border of attracting mass ..................................... 46

X. Slope of disturbed surface............................................ 47

28. Derivation of expressions for slope of disturbed surface........... 47

29. Failure of these expressions in special case of slope at border of

distui'bing mass ............................................... 47

30. Derivation of expression for slope at border of disturbing mass.... 48

XI. Disturbed center of gravity of earth .................................. 51

31. Centers of surfaces of reference appropriate for different purposes; 

derivation of modifications of preceding formulas wlieii disturbed center of gravity is center of surface of reference....... 51

XII. Equations of disturbed surface when disturbing mass is of variable

thickness..................................................... 52

32. Desirability of extending the investigation to more complex disturbing masses................................................ 52

33. Derivation of expression for effect of any mass symmetrically disposed about a radial axis, and application to a class of mass- 

*- forms.................. ....................................... 53

34. Evaluation of a definite integral needed in applications of sequel. 55

35. Elevation of disturbed surface at the center, at the border, aud 

at the point 180° from the center of the disturbing mass in the 

case of the above class of mass-forms.......................... 56

36. Slope of disturbed surface ................. ...................... 56

37. Effect of re-arranged free water.......................... ........ 56

38. Remark on a property of certain formulas of this article.......... 5 -J

B. APPLICATIONS.

XIII. Relative positions of level or equipotontial surfaces in a lake basin.... 58

39. Solution of problem stated in section 2 («)....................... 53

(90)
CONTENTS. - 7

Page.

40. Illustrative numerical example .................................. 5'J

41. Inference from preceding solution................................ GO

XIV. Variations in sea level attributable to continental glaciers or ice caps.. 60

42. Statement of problem, and brief consideration of the first of two

difficulties .................................................... CO

43. Consideration of second difficulty................................ 61

44. Data assumed for calculation .................................... 61

45. Definition of forms of assumed masses.... .1.. ...<................ Gl

46. Information as to actual forms of the ice caps, and reasons for considering assumed forms adequate .............................. 62

47. Computation of the volumes of the assumed masses and equivalent 

lowering of sea level..........................................

Table of results.................................................

48. Remark on the magnitudes of the masses of the assumed ice caps in

comparison with the earth's mass.............................. 65

49. Computation of position and slope of disturbed surface........... 65

Table of results...... ............................................ 66

50. Estimate of the effect of the re-arranged free water, and discussion

of results ............. .......... ...................... ........ 67

51. Minimum thicknesses of ice masses of varying radial extent, requisite to produce average slopes of 5 feet per mile within 1° of their 

borders........................... ............................ 68

Table of results............ ..................................... 68

52. Variations in sea level due to alternation of glaciation at the poles 69 

Table of results.................... ............................. 70

Graphical representation ......................................... 70

X A'. Historical note....................................................... 71

53. Reference to discussions and investigations of previous writers on

the effect of the glacial accumulation in disturbing the sea level - 71

54. Investigations of Archdeacon Pratt.................. ............ 71

55. Numerical calculations of Pratt.................................. 72

56. Test of the correctness of Pratt's formula ........................ 73

57. Investigations of Mr. D. D. Heath............................... 74

58. Verification of a numerical example in Heath's work ............. 75

59. Heath's criticism of Croll and Pratt.............................. 75

60. Contribution of Sir William Thomson ; proofs of Thomson's formula ................ ........................................ 76

61. Verification of his numerical example............................ 76

62. Remarks on the results obtained by different writers and tabular

statement of the data employed by them....................... 78

Table of data used .............................................. 79

XVI. Variations in sea level attributable to continental masses*..... .... .... 79

63. Two hypotheses relative to the nature of the earth's crust.... .... 79

64. Assumptions adopted iu accordance with first hypothesis ......... 79

65. Data for and methods of computation............................ 80

Table of results....... ...... ................ ...... .............. 81

66. Graphical representation of results .................. ...... ...... 82

67. Elevation of disturbed surface at the border of the continent..... 82

68. Remarks on the resultant action of the continents ............... 82

69. Deflections of the plumb-line along the border of the continent... 83

70. Consideration of the effect on the sea level of the continents under

the conditions of the second hypothesis........................ 83

71. Deflection of the plumb-line...................................... 85

XVII. List of authors consulted ......................... .................... 85

72. Authors, titles of their works, and datesof publication........... 85,86

(91)

KEY TO MATHEMATICAL SYMBOLS.

[The page numbers refer to pages where the symbols arc first introduced and explained in tbo course

of the investigation.]

Page. 

a =a special value of the radius-vector of tbo sea surface used

in the investigations of Archdeacon Pratt................ 71

a0=the equatorial semi-axis of the earth's spheroid or thatoblato

spheroid which coincides closely with the sea surface..... 10

£=a certain definite integral ................................. 55

6=sini/? .................................................... 4:5

&or=polar semi-axis of the earth's spheroid..................... . 19

Z»o=the greatest value of b when it is variable.................. 55

Ci=a constant used temporarily for brevity.................... 19

C'2=a constant used temporarily for brevity.................... 20

c =2 r0 sin /?.... .............................................. 50

c, (I, e . . .= symbols for constant coefficients used only temporarily..... 44

c0 =a constant used temporarily for brevity.................... 36

.D=the distance between the attracted and any attracting point. 21 

7^-(|3)=a definite integral, a function of the angle /?, i being any

positive integer ....... .................................. 33

f-t (cos 0)=/i (/£) a polar harmonic, a function of the angle /?, i being any positive integer.............................................. 33

(j =the velocity increment due to the earth's attraction on bodies 

outside of and near to its surface; y is about 32 feet if the 

mean solar second is the unit of time .................... 20

<7i,<72 , #3 . . . =symbols for functions of b or the angle /3 .................. 44

7) =the thickness of the attracting mass measured along a radius

of the earth............................................. 22

/i 0=the maximum value of /< when it is variable............... 53

/j 1( 7i 2=height above sea level to which a stratum is supposed to bo

raised .................................................. 83

I, Ji, J2=certaiu definite integrals, / being used for either I} or J2 when

it is not necessary to particularize........................ 24

i=any positive integer....................................... 32

Jo, «/i> -Tiz =symbols for certain definite integrals ...................... 32

Jo, ~J\t Ji' =symbols for certain definite integrals ...................... 32

 Jo", J\"t Jv" =symbols for certain definite integrals ...................... 32

ji>fa,J3 ^constants which are functions of the angle a............... 55

7t- i, fc2 , fo =constauts which are functions of the angle /3 ............... 45

Z=the distance of any point of the sea level from the earth's^

axis of rotation ......................................... 18

loge=Naperiau logarithm ....................................... 32

,M=the mass of the earth...................................... 19

m=a disturbing or attracting" mass situated anywhere with respect to the earth, and especially on or near the earth's 

surface ........................................... ...... 19

^1,^2,^3 : =symbols for certain numerical quantities................... 65

«=any positive integer....................................... 44

(03) 9
10 KEY TO MATHEMATICAL SYMBOLS.

Pago.

P =the potential of the earth's mass with respect to a particle 

or unit mass on its surface. P is the sum of every masselement of the earth divided by its distance from the attracted particle......................................... 18

P0,P\,Pv . . . =synibols for spherical harmonics or Laplace's coefficients... 31

p=an auxiliary angle and subject variable of a definite integral . 23

Qi, <2-2=certain factors used temporarily for brevity ................ 29

g=an auxiliary angle .................. ...................... 23

r =t.he radius-vector of any element of the attracting mass.... 21

n>=the radius of a sphere of equal volume with the earth's

spheroid. It is the radius of the sphere of reference.-... 19 

r'=the-radius-vector of the attracted point or particle......... 21

S, Si=sums of certain series ..................................... 56

s, «o=auxiliary angles used temporarily ......................... 23

/, ti, /2=auxiliary quantities, t being the subject of a definite integration, and ti and fe the limits of *...... .................. 38,39

ti, <2=thicknesses of strata...................................... 83,84

J70=a constant iised temporarily............................... 36

«=the excess of r over ?*o.-----..----..-----.----..----..----' 21

F =the potential of the disturbing mass TO. .................... 19

F0=a constant. It is the value of F along the line of intersection of the disturbed and- undisturbed surfaces........... 20

v =the elevation or depression of the disturbed sea surface relative to the undisturbed sea surface. It is the same as 

(r'-r0) .---..----...............-...-.......---...---.. 20

i?o=a constant used temporarily............................... 20

v} , v2, f3=certain special values of v ................................. 41

 )/=the elevation or depression of the sea surface relative to a 

spherical surface concentric with the disturbed center of 

gravity of the earth .................. .................. 52

v" =elevation or depression of the disturbed sea surface relative 

to the undisturbed surface when the disturbing mass is 

of variable thickness ................................... 53

Vi", v.2", r3"=certain special values of v"................................ 56

rft=:elevation of disturbed surface at a point whose angular distance from the center of the disturbing mass is a......... TO

-u'=sin ^a/sin |/3-............................................. 43

.X==a definite integral......................................... 46

x, y, #,=rectangular Cartesian co-ordinates ........................ 21

x', y', s'=rectangular Cartesian co-ordinates ........................ 21

Fo, Fi, F2 , . . . ^certain spherical harmonics or Laplace's coefficients ....... 36

ZQ, Zi, Z.2 , . . . =certain spherical harmonics or Laplace's coefficients ....... ~ 36

a=the angular distance of any point of the disturbed sea surface from the center of the disturbing mass. Jt is the 

angle at the center of the sphere of reference between a 

line drawn to any point of the disturbed surface and a 

line drawn to the center of the disturbing mass.......... 23

[3 =the angular radius of the disturbing mass or the angle at 

the center of the sphere of reference between a line drawn 

to the center of the disturbing mass and one drawn to its 

border. /? is used in this sense with reference to masses 

of uniform thickness.................................... 23

/?o=the greatest value of /3 when it is variable; i. e., /30 is the angular radius of a mass whose thickness is variable....... 53

/3i, /?2=]iniits of /? when it is a subject of integration.....'......... 53

(94)
KEY TO MATHEMATICAL SYMBOLS. 11

Tage.

/?'=the angular radius of a conical ring or annulua and a subject of integration .................................. .... 51

y,, y.2=auxiliary angles and subject variables iu the definite integrals Ji and J2 .......................................... 24

A V =a finite change in or increment to V...... ................... 36

A P'0=a finite change in or increment to F0 . ...................... 36

4v=a, finite change in or increment to v........................ 36

4p=a finite change in or increment to p ........................ 58

/ v;=an ordinate and subject variable .......................... 49

0, 0'=angles corresponding to polar distances.................... 21

A, A'=angles corresponding to longitudes ........................ 21

7/=cosec ^a ...... ............................................ 45

£=an ordinate and subject variable .......................... 49

;r=the ratio of the diameter to the circumference of a circle

=3.14159+ ............................................. 19

p =ihe density of the attracting or disturbing mass m......... .19

p =the mean density of the earth=5.5, as used in this paper.... 19

pw =the density of sea water=l,as used in this paper........... 36

pi, p2=the densities of strata whose thicknesses are t\ and t2 , and

heights above sea level 7ii aud 7i 3 ......................... 83,84

' 6 =the displacement of the earth's center of gravity, caused by

a superficial mass having a circular border............... 51

tfi=the displacement of the earth's center of gravity due to the 

shifting of a hemispherical meniscus from one hemisphere 

to the opposite one..'................./.................. 78

r=the thickness of a spherical shell of equal volume with*assumed ice mass ......................................... 04

(p({S) ihe variable thickness of an attracting or disturbing mass, /3 

being the angular distance of any part of the mass from 

its center............................................... 53

^=an auxiliary angle used temporarily only .................. 21

63=:the angular velocity of the earth about its axis ............ 18

<a=the ratio of the area of the ocean to the area of the earth's

surface in a case assumed by Sir W. Thomson ............ 76

(95)

LETTER OF TRANSMITTAL.

DEPARTMENT OF THE INTERIOR,

TT. S. GEOLOGICAL SURVEY, 

Washington, D. C., May 31,1887.

SIR : I have the honor to transmit herewith the results of certain 

investigations, which may be broadly designated as relating to the form 

and position of the sea level. These investigations were begun in part 

previous to my connection with the G-eological Survey, but they were 

taken up again in 1885, with your approval, at the request of Mr. G. K. 

Gilbert and Prof. T. C. Ohamberlin, for solutions of some special problems which arose in their geological researches. The work has been 

prosecuted simultaneously with other lines of office and field work. It 

reached its present form substantially, however, more than a year ago; 

and the principal numerical results of the discussion of Professor Guamberlin's problem are incorporated with his paper on The Driftless 

Area, in the Sixth Annual Report. The purely mathematical features 

of the paper have been published also in the Annals of Mathematics, 

Nos. 5 and 6, vol. 2. and No. 1, vol. 3. I have delayed offering the 

complete manuscript for publication up to this time in order that I 

might give it a careful revision and check all the more important form-, 

ulas by independent processes of derivation.

The questions treated in this paper are for the most part necessarily 

somewhat mathematical. They are, however, fundamental questions in 

geophysics, and although the mathematical form of presentation has 

been followed throughout, an attempt has been made to state the end 

results and formulas in such a way that they may be understood and 

used with safety by those who may not care to follow the details of the 

analysis. For the benefit of such readers a key to the mathematical 

symbols employed is given in addition to the list of contents and general index.

While the analysis of this investigation was designed especially to 

solve the particular problems of Messrs. Gilbert and Chainberliii, it has 

not been confined to those problems, but has been adapted to the entire 

class of problems to which they belong. It is hoped, therefore, that 

the results of the paper will be of interest and value to geodesists and 

mathematicians as well as to geologists.

Very respectfully, your obedient servant,

R. S. WOODWARD. 

Hon. J. W. POWELL,

Director If. S. Geological Survey.

(97) . 13

ON THE FORM AND POSITION OF THE SEA LEVEL

BY E. S. WOODWARD.

I. INTRODUCTION.

1. The problem, of the form and dimensions of the sea level surface 

of the earth has been one of peculiar difficulty. The combined efforts 

of the ablest mathematicians of the past two centuries, supplemented 

by the most laborious and costly geodetic measurements have yielded 

us the first approximation only to the complete solution. Fortunately 

this first approximation is exceedingly close. It assigns to the sea level 

a form which differs but slightly from that of an oblate spheroid, whose 

major and minor semi-axes are about 20,926,000 and 20,855,000 English 

feet, respectively. This spheroid, or reference ellipsoid, as it is sometimes called, has its minor axis coincident with the earth's axis of rotation and is usually regarded as sensibly fixed in position and dimensions. With respect to it the actual sea surface or geoid must be 

imagined to lie partly above and partly below by small but unknown 

amounts, the determination of which, if possible, will constitute a 

second approximation to the figure of the earth. For many if not most 

of the applications of science the reference ellipsoid suffices; the first 

approximation is nearly enough correct. But geodesy, on the one hand, 

has attained such a degree of perfection in precise measurement that 

the discrepancies now brought to light in some of its operations must be 

attributed largely if not chiefly to defects in theory. These discrepancies must be explained before any considerable advance can be expected 

in our knowledge of the figure of the earth along the present lines of investigation. Their true explanation is apparently intimately connected 

with the form of the geoid, and it is to the study of the form, therefore, 

rather than to the determination of the dimensions of the geoid that we 

may look for future progress in geodesy. Geology, on the other hand, 

has raised many questions relative not only to the form, position, and 

fixity of the geoid properr but also with respect to the allied equipoten-

(99) 15
16 FORM AND POSITION OF THE SEA LEVEL. [BULL. 48.

tial surfaces of isolated bodies of water at higher or lower levels. It is 

found, for example, in geological investigations, that the shore lines of 

extinct seas do'not always coincide with existing level lines, but often 

cross them at decided angles, or that the water level lines traced on 

islands in such extinct seas differ in elevation from contemporaneous 

lines traced on their distant shores. Aside from the changes which 

may have been due in these cases to subsidence or upheaval, the question may be raised whether such slopes or differences in elevation relative to present level lines may not have been caused by adjacent 

attracting masses, which have since disappeared, like the ice mass of 

the glacial epoch, or in a lake basin by the presence of the water itself. 

Correct and complete answers to such questions require a knowledge 

of the existing geoid and of the causes which may have produced secular variations in its form and position. 

At present it is by no means clear how any specially extensive additions to our information concerning the more minute features of the sea 

surface are to be obtained. It may even be doubted whether we hare 

not reached a practical limit in the first approximation; whether in fact 

the distribution of matter within the earth's surface is not so irregular 

as to preclude gaining anything more than an empirical formula for the 

deviations of the geoid from the ellipsoid of reference. It seems probable, however, that the forces producing these deviation's have their 

seat in a comparatively thin terrestrial crust resting on a fluid or plastic 

substratum (or nucleus), or that such was the antecedent condition of 

the earth, and that our failure to perceive the relations of the crust to 

the substratum is the chief obstacle to improvement.

In the absence of a complete rational theory the best evidence which 

analysis can bring to bear on questions pertaining to the geoid is largely 

of a negative character. The effects which would result under certain 

conditions can be computed, but it is not always possible to prove that 

those conditions accord with the actual facts. Investigation must 

proceed to some extent upon doubtful postulates, and computations 

must be made from uncertain data. But notwithstanding this limitation on the calculations we are about to consider, they will generally 

possess a value in excluding or confirming hypotheses, or in furnishing 

limiting values for the effects of observed causes.

2. A considerable class of problems concerning the sea level is that 

iii which the attracting or disturbing mass is symmetrically disposed 

about a radius of the earth's surface, and is situated on or near the 

surface. As examples of this class we may adduce the two following, 

which led to this investigation :

(a) Given the dimensions of a lake basin having a circular border. 

When the lake was full of water it left a trace of its surface along the 

border and on an island at the center of the basin. After the water 

had disappeared a line of spirit levels was run between the water

(100)
WOODWAED.] RESUME OF RESULTS. 17

on the island and that on the border j what difference in altitude should 

have been found ? l

(6) Assuming the accumulation of ice in glacial times to have been 

in the shape of a spherical stratum bounded by a circle, or some sort of 

meniscus symmetrical about an axis, and that the earth's crust did not 

yield under the weight of the ice, .what were the resulting distortions 

in the sea level? 2

It will be seen that these problems are essentially the same. They are 

substantially identical also with the problem of the effect of continental 

masses on the sea level, since the continents may be represented, approximately at least, as spherical strata having circular borders, or as 

masses of meniscoid shape. *. o 

3. The following paper is devoted to the investigation and discussion 

of this class of problems. An attempt has been made to develop the 

theory of their solution so far as is necessary to render practicable the 

numerical evaluation of the characteristic effects of the disturbing mass 

in any special case. In Articles II to XI the theory of the effect of a 

mass in the shape of a spherical stratum having a circular border and 

uniform thickness is worked out with considerable detail. The only restrictions imposed on this mass are that its density is uniform, and that 

the ratio of its thickness to the earth's radius may be neglected in comparison with unity. Expressions for the potential of the disturbing mass 

at any point of the disturbed surface are derived in terms of a definite 

integral and in terms of spherical harmonics; and the degree of approximation of these expressions is investigated. Equations to the disturbed surface are assigned for the case in which the effect of the rearranged free water is considered, as well as for the case in which that 

effect is neglected. The disturbance in the former case is shown to be 

equal to that in the latter, which is expressed in compact integral form, 

plus a rapidly converging series of additive terms.

1 This problem was proposed by my colleague, Mr. G. K. Gilbert, to tbe mathematical 

section of the Washington Philosophical Society, February, 1884. In his geological 

investigations within the area of the Quaternary sea known as Lake Bonneville, Mr. 

Gilbert has found traces of the central portions of the ancient lake surface to be more 

than 100 feet higher than the traces of the contemporaneous surface at its margin. 

A complete consideration of the effects of the causes which might contribute to this 

distortion requires, obviously, a numerical evaluation of the depression of the level 

surfaces within the area, due to the removal of the water.

A solution of the problem was given by the writer before the above-named society 

in March, 1884, and a more complete discussion will be found in sections 39-41.

2 To what extent the form and position of the sea level may be modified by the mere 

attraction of glacial masses is a question which has been much discussed by geologists. 

It was proposed to the writer by Prof. T. C. Chamberlin, geologist in charge of the 

division of glacial geology, U. S. Geological Survey. The question is considered at 

some length in sections 42-52, and a review of the work of the more prominent 

mathematicians who have discussed the problem is given in sections 53-62. The 

principal numerical results of the writer's investigations are given in Professor 

Chamberlin's paper on Th« Driftless Area, in the Sixth Annual Report of the U. S. 

Geological Survey. 

Bull. 48 2 (101)
18 FORM AND POSITION OF THE SEA LEVEL. [BULL. 48.

In Article Xtl the investigation is extended so as to assign the effect 

of any mass of uniform density having a symmetrical distribution about 

a radius of the earth's surface. Particular attention is paid to a class 

of masses whose shapes are assigned by a formula which represents 

fairly well the mass features of the problems (a) and (6) above.

Under the head of applications, Articles XLII to XVI, the characteristic properties of the equipotential surfaces in a lake basin are first 

considered. Then the variations in sea level attributable to continental 

glaciers or ice caps are discussed at some length. The angular radial 

extent of the ice mass is, for the most of the discussion, assumed to be 

38°, for the reason that this is the 'extent of a mass of nearly uniform 

thickness, which would produce the maximum upheaval of the water 

along its border. The external shapes of the various masses, their volumes, and the distortions of the sea surface attributable to them are 

given in detail. The minimum thicknesses of ice masses of varying 

angular radial extent, requisite to produce average slopes of five feet 

per mile within one degree (69 miles) of their borders, and the extent of 

variation in sea level on the hypothesis of an alternation of glaciation 

at the poles of the earth, are also worked out.

In the historical note of Article XV, the allied investigations of Archdeacon Pratt, Mr. D. D. Heath, and Sir William Thomson, on the problem of glacial submergence are reviewed. The special cases they have 

considered are shown to be easily derived from the general formula of 

Article XII.

Finally, in Article XVI, a brief discussion of the effect of continental 

masses in distorting the sea level is given. It is shown that according 

as the continents are or are not superficial masses unbalanced in their 

attractive effects, the sea surface must be very irregular or deviate only 

by minute quantities from the ellipsoidal form. It is also shown that 

although a continent whose radial element masses are in a condition 

bordering on hydrostatic equilibrium would produce but slight disturbances in the position of the sea level, it might nevertheless cause a»considerable slope of the sea surface, or deflection of the plumb line along 

its border.

A. THEORY. 

II. MATHEMATICAL STATEMENT OF PEOBLEM.

4. The solution of the general problem outlined in the preceding section depends on the principle of hydrostatics that the potential of the 

forces producing a liquid surface in equilibrium has a constant value 

for all points of that surface. In the case of the earth, if the potential 

of all the attractive forces acting on a unit mass at any point of the sea 

surface be denoted by P, the distance of the point from the earth's axis 

of rotation by I, and the velocity of rotation by c», the form of the surface will be completely defined by the equation.

(102)
WOODWARD.] MATHEMATICAL STATEMENT OF PROBLEM. 19

2=a constant. (1)

The exact value of P in this equation is a complicated function of the 

densities of the element particles of the earth and of the co-ordinates of 

those particles and the attracted point. For the present purposes, however, it will be sufficient to consider P due to a centrobaric sphere of 

equal mass and volume with the earth and concentric with the earth's 

center of gravity. Since we shall only consider relative positions of 

any point on the sea surface, the potential due to centrifugal force, which 

is represented by the secoud.term in (1), may be neglected.

5. If OQ and &0 denote the equatorial 'and polar semi-axes, respectively, 

of the earth's ellipsoid, and r0 the radius of the sphere just referred to;

*o= X2&o. (2)

Using Clarke's values1 of a0 and &0 we have

a0 =209260G2 English feet, 

&0 =20855121 English feet, 

r0 =20902391 English feet, 

log r0=7.32020.

The surface of the sphere thus defined may be regarded as the surface assumed by a thin film of sea water covering a nucleus whose mass, 

plus the mass of the film, equals the earth's mass. We shall call this 

ideal surface the undisturbed surface. With respect to it the real surface of the earth lies partly without and partly within; but so far as 

small relative changes in sea level are concerned it is practically immaterial whether we refer to the actual closely spheroidal surface or to 

the simpler spherical one. 

6. Let .

lT=inass of the earth, 

pm =inean density of earth. 

Then,

M=±7ir,?pM (3)

and the equation to the undisturbed surface is

^=Wp.= 0i, ' (4) 

r0 o

C\ being a constant.

Suppose, now, a new mass, m, of density p (positive or negative) be 

placed in any fixed position relatively to the undisturbed surface. The 

resulting sea surface will then differ from that defined by (4). To determine this difference let V be the potential of the disturbing mass m

1 Comparisons of Standards of Length, made at the Ordnance Survey Office, Southampton, England, by Capt. A.. R. Clarke, E. E. Published by order of the secretary 

of state for war, 1866,

(103)
20 FORM AND POSITION OF THE SEA LEVEL. [BULL. 48.

at any point of the disturbed surface, and let v denote the elevation or 

depression of this point relative to the undisturbed surface. The equation to the disturbed surface will then be

Ttr

F=02 , a constant. (5)

The difference of this and (4) to terms of the first order inclusive in

v is

whence, putting

Since M/r02=g, the velocity increment at the earth's » surface due to 

the earth's attraction, (6) may be written

Vo in the last two equations is the value of V w.heii v=Q, or the value 

of V along the line of intersection of the disturbed and undisturbed 

surfaces. If we put

This equation represents the elevation of the disturbed surface above 

a spherical surface of equal potential, whose value is

 = ^2j

since the difference between this and (5) gives (7).

The constant F0 may be determined from the obvious condition that 

the disturbed and undisturbed surfaces must contain equal volumes.

It is evident that the equations just derived will hold true if the 

mass m be a part of the earth's mass, so long as the ratio m/M may be 

neglected relatively to unity. Thus, in the problems we shall consider, 

m may represent the mass of a continent, the deficiency in mass of a 

lake or lake basin, or the ice mass of the glacial epoch.

(104)
WOODWARD.] POTENTIAL OF DISTURBING . MASS. 21

III. EVALUATION OP POTENTIAL V DISTURBING- MASS ON SURFACE 

OP EARTH, OP UNIFORM THICKNESS AND DENSITY, AND WITH 

CIRCULAR BORDER,

7. The next step in the solution requires the determination of the 

potential Fof the attracting mass for any point of the disturbed surface, whether without or within the circle which we have assumed to 

define the boundary of the mass. Although the nature of the mass may 

be such as to prevent the water from permeating it freely, the surface 

the water would take if not so restricted is an essential part of the 

disturbed surface.

In order to derive an expression for V, let the rectangular and polar 

co-ordinates of any point of the attracting mass be defined by the usual 

relations, viz :

x=r coS'0 cos A,

y=r cos 0 sin A, 

s=r sin 0,

in which 6 and A correspond to polar distance and longitude, respectively, the position of the origin being arbitrary. With reference to 

the same origin, let the co-ordinates of the attracted point on the sea 

surface be 

x'=r' cos 0' cos A',

y'=r' cos 0' sin A',

z'=r' sin 0'.

If D denote the distance between the attracting and attracted points 

and

cos ?/>=cos 0 cos 0'+sin 0 sin 0' cos (A A'), (8)

D2=r2+r'2 -2rr' cos ^=(r-r')2 +4n-' sin2 £ . (9)

2

The volume "element of the attracting mass is 

dxdydz=r2dr sin OdQdX.

Hence, if p denote the density of the attracting mass, a general expression for the required potential is

We must now eval uate this integral. Taking the center of the sphere 

of reference as the origin of co-ordinates, let

and (11)

r'=r0 -kv, 

(105)
22 FORM AND POSITION OF THE SEA LEVEL. [BULL. 48.

in which u and v are small quantities relative to r0 , v being the same 

of course as defined by equation (6). Premising what will be proved in

Article IV, namely, that we may neglect quantities of the order , -,

and ( ^n^ ) equation (9) gives 

V ^o s '

D=2r0 sin -. (12) 

From the first of,equations (11)

and v (13)

to terms of the order - .

r0

As to the magnitude of the quantities neglected, it may be remarked 

in passing that r0 is in round numbers 2l,000,.000 feet (see sections), 

while u and v may be restricted to values less than 100,000 feet; so 

that the fractions neglected will not exceed o-^ .

Without loss of generality we may assume the line from which 6 and 

0' are reckoned to pass through the attracted point, and the* plane from 

which A and A; are reckoned to pass through the attracted point and 

the center of the attracting mass. In this case 0'=0 and A'=0, and (S) 

gives $-=6.

By means of this relation and the equivalents in (12) and (1.3) the integral in (10) becomes

V=r0p C C Cdu cos ~ d0d\. . (14)

If the uniform thickness of the attracting mass be denoted by A, the 

limits of u in (14) will be 0 and h. Let the limits of 6, which are obviously functions of A, be denoted by 61 and 6Z . The limits of A are 

evidently equal in magnitude but of opposite signs. Hence we have

V=2roP Fdu Pcos-fd0 (*&*.=& Jip /Ysin ~- sin ~

J o J e, * J o J o\ ^ *

8. To complete the evaluation of (15) it will be convenient to change 

variables. Consider the spherical triangles formed by the attracted and 

attracting points, the center of the attracting mass, and the points in 

which the arc Q cuts the circle bounding the mass. Thus in Figs. 1 and 

2, let P be the attracting and A the attracted points, G the center of 

 the attracting mass, and BDGr the bounding circle. Then

(106)
WOODWAHU.] POTENTIAL OF DISTURBING MASS. 

Draw CE perpendicular to AB and put

AC=a, BC=/3, 

BE=s0) 

AE=q,

From either figure

6=q+s, 

whence

° sin -siu .=2 cos sin ° .

2 . 2i 2i 2i

The right-angled spherical triapgles of either figure give 

cos =

cos p cosp

PIG. 1.

The first two of (17) give

PIG. 2.

COS

23

(16)

(17)

cos jp

whence

cos

COSJ?

From the last of (17)

 COS2

(18)

(19)

Now, the last of equations (17) and the diagrams show that the limits 

of p, corresponding to the limits of A, are 0 and a or 0 and /3f according 

as the attracted point is within or without the circle bounding the at-

' (107)
24 FORM AND POSITION OF THE SEA LEVEL. [BULL. 48.

tractiug mass. Hence, if we denote the potentials in the two cases by 

Vi and F2 , respectively, the equivalents in (15), (16), (18), and (19) give

P

I

I

I 

J

i

/COSfl COS/JV ,1 .__. ( ± '- } dp. (20) 

\cosp cos aj -^ v ' 

o

(21)'

9. The integrals in these equations are in general elliptics of the third 

species. They may be evaluated by the usual processes applicable to 

elliptics, by series, or by mechanical quadrature.

The integral in (20) presents some apparent difficulty, since the element function is infinite at the upper limit, except when a=/3. Again, 

in case a=0, this integral assumes the anomalous form

the value of which is it sin '-. as may be easily verified by means of (15),

2i

(16), and (18). These peculiar features may be removed by the following change of variables, which secures the same constant limits for both 

(20) and (21). 

For brevity put

- 

\cosp cos r 

["

I

J o

(*

|

I

I

Jo

| /cos p cos/?

( £ '-

o'V cos p cos a

Then, observing that .

. sin2 |-sin2 | 

cosp cos yg_ 2 ____ 2

put in /i

sin^=sin^ sin

and in J2

. V . 0 . QI -r\ -*L^~ Q1 Tl ]_ QlTl

.'108)
WOODWARD.] SPECIAL VALUES OF INTEGRALS I, AND

These give

2 sin ~ cos

and

2 sin cos

25

and the limits for both yi and yz are 0 and ^. Therefore

PI

sin**

(1 -sin2 * sin2

2 sin2 '- cos2

sin2

'

(24)

-,. (25)

7*

10. Some special values ot tffe integrals (22) to (25) and the corresponding potentials (20) and (21) are worth deriving. These values are:

(a) For a point of the disturbed surface at the center of the disturbing mass, «=0, and (24) gives

7T

!=:2sin^ f 2 dn=

2 Jo

and the corresponding value of the potential is

(26)

 (27)

a

(7>) For a point at the border of the disturbing mass, <*=/?, and hence 

from (24) and (25)

7T

r /5 , yz=

/!=J2=2 arc sin sin ii sin ^ "=A ' (28)

-J y ;Q

(109)
26 FORM AND POSITION OF THE SEA LEVEL. [BULL. 48.

a result which is reached more readily from (22) or (23). The corresponding value of the potential is

F1 =F2 =4r0 7i/o/?. (29)

(c) For a point of the disturbed surface 180° from the center of the 

disturbing mass, a=n, and (25) gives

1 sin2 ' cos2 y<i&yi

2_______-9

= ^r 2 cos 4-arc tan tan( cos-^-tan/2 ) J 2 ^ L \ 2 yj

f, (30)

and the potential is * * 

F2 =8r0 7&p7r sin2 j. (31)

(tZ) Suppose fi=n\ in other words, let tbe attracting mass cover the 

whole sphere. Then (24) gives

Ji=*, (32)

and we have the following well-known approximate value for the potential of a spherical shell for a point on its interior or exterior surface1 , 

viz:

4f0 /i/37T. ' (33)

This result follows also from (27) or (29) if we make ft=n.

IV. DEGREE OF APPROXIMATION OF THE EXPRESSIONS FOR THE

POTENTIAL 7.

11. In deriving the expressions (20) and (21) for the potential of the 

disturbing mass, it was assumed that a sufficient degree of approximation is attained if quantities of the orders -, -, and upwards are

neglected. The grounds of this assumption need to be examined with 

some care. For this purpose we shall derive the exact expressions, in 

form at least, for the potential of the disturbing mass at its center, at 

its border, and at 180° from its center. A comparison of these exact 

values with the approximate values given by (27), (29), and (31) will 

show the order of approximation of (20) and (21).

1 See equation 34. 

(110)
WOODWARD.] DEGREE OF APPROXIMATION ATTAINED. ' 27

We will first write down the expression Jtbr the potential, of a complete spherical shell, of uniform thickness and density, for a point within 

its bounding surfaces. This expression will be useful as a check on 

formulas relating to partial shells.

Let the radius of the interior surface of the complete shell be r0 , the 

uniform thickness h, the uniform density p, and the distance of the attracted point from the center of the shell rQ +v.

Then the potential is 1

-- /VA\ (34)

12. Let the notation be the same as that used heretofore; i. e., let r 

be the radius- vector of any point of the attracting mass, r' the radiusvector of any point of the disturbed surface, p the density, h the thickness, and ft the angular extent of the mass; and 6 the angular distance 

between the attracted and attracting points. For points of the dis- 

, turbed surface lying above the undisturbed surface, r will be less than 

rl =r() -\-v over the range r0 to r', and r will be greater than r' over the 

range r' to r0+h. Bearing these facts in mind it follows that the exact 

Value of the potential of the disturbing mass for the point where its 

axis pierces the disturbed surface is

rr « 7=2ft*

sin Ode

r0+v U 0 ^

J

See Price'8 Calculus, vol. 3, p. 299. 

(Ill)
28 FORM AND POSITION OF THE SEA LEVEL. fBULL. 48.

Since

,_./.,, (r-r1 )2

2r sin zf 1+

2i

(r r')2

Sr'r sin2 ft

1-r_ri

we find by expansion, integration, and reduction, to terms of the first 

order inclusive,

3h sin ~ li

1+

v{ 1 sin

sn

. (35)

If we make (3=7t this expression agrees with (34) to terms of the 

second order.

Equation (35). it will be observed, differs from (27) by certain terms 

which must be small unless li and v are very large. In one of the most 

important applications discussed in the sequel, ^=10,000 feet, 0=3,000 

feet, and /?=3S°. With these values, since r0 is in round numbers 

21,000,000 feet, the quantity within the parentheses of (35) differs from 

unity by less than y^o. I1' /5=CO°, which is (see section 23) the anT 

gular extent of mass required to produce the maximum elevation of the 

disturbed surface at the center of the mass, the quantity within the 

brackets of (35) exceeds unity by less than goVo* lisiu§ ^e above values 

of li and v. 

13. Similarly, if the attracted point be at the border of the attracting 

mass, the exact value of the potential is

Ldr r'

'j

/

»o+ v XT' 

r / (r' r} dr p7t / 

T' / 

r. t' ,

(112)
WOODWARD.] DEGREE OF APPROXIMATION ATTAINED. 29 

in which

sn 2 l-sin^siaU*

Introducing this value, the first of the above integrals relative to A 

becomes

IT

Since the numerator of the element function in this integral is greater 

an sin ft cos /I andless than sin. 

tegral lies between (r'-\-r}ft and

than sin ft cos /I andless than sin. ft cos A 4 ^jj- cos/?, the value of the inSuppose the-exact value of this integral is

Likewise, represent the exact value of the second integral relative to 

A by

Then to terms of the first order inclusive the above expression for 

the potential becomes

The first term of this agrees with (29). 

The quantities Qi and Q% lie between

A ^ 0 and

"2

&\

ft I Vl sin2 y6fsin2/l'

n When ft= rt , or when the attracting stratum covers a whole hemi- A

sphere, Q} = Q2 =Q, and (36) becomes

(36') v '

(113)
30 FORM AND POSITION OF THE SEA LEVEL. IUULL. 48.

This agrees, as it should, to terms of the second order with the half of 

(34). For 7i=10,000 feet and ^=3,000 feet the quantity in the parenthesis exceeds unity by less than T-^O-Q. When /?<' ?,, Qi and Q2 wi]] exA

ceed 0, and the degree of approximation of (36) will be somewhat higher 

than that of (36').

14. For a point of the disturbed surface 180° from the center of the 

disturbing mass the exact value of the potential is

=2p7t: 9 .J I r2 dr ,. - 

Vr2+r'2 +2rr' cos 0

=lp7t I (r-fr' \/r2+r'2 -\-'2rr' cos/?) «

t/»o . ^

When

(r r')2 <4rr'cos2 {- ,

2

which is the only case we need consider,

_ ______________ ft __ C ' (;.__ r/\2

-v/r2_|_ r/2_(_2rr/ cos y^ 2 cos 7^ v/rr' ^ 1+ - ;\ 

\ 8rr' cos2 S-

.arco.

Hence, expanding, integrating, and reducing, there results to terms of 

the first order inclusive.

(37)

the first term of which agrees with (31). ' Using the values r0 =21,000,000 

feet, 7&=10,000 feet, and v= 1,000 feet (v being here intrinsically negative), the factor in the brackets of (37) exceeds unity by ^

Y. DEVELOPMENT OF POTENTIAL V IN SERIES OF SPHERICAL

HARMONICS.

15. The preceding expressions for the potential of the attracting mass, 

namely, T7, as defined by equations (20) and (21), are sufficient for most 

of the applications to be considered in the sequel. They possess the 

obvious advantage of a compact integral form. For some purposes, 

however, it will be desirable to have V expressed in a series of spherical harmonics or Laplace's functions. We may thereby arrive at equa-

(1U)
WOODWARD.] HARMONIC DEVELOPMENT OF POTENTIAL. 31

tions (20) and (21) by a process differing from that followed in Article III 

and establish a useful harmonic development of the elliptic integrals 

Ji and I2 of equations (22) to (25).

16. Expressions fulfilling the present .requirements may be derived 

from equation (10) by expanding D~l in a series of ascending powers of

/y ij*

 and . Thus from equation (9) we have

+ . . . ] when r<r',

v\ 2 n

^) + . . ] when r>r>.

In these equations P0) -Pii -?2, etc., are Laplace's coefficients of the 

zero, first, second, etc., order, respectively. They are functions of the 

angular coordinates 6, 6', .A, and A' only.

Taking, now, the center of the sphere of reference as the origin of coordinates, and supposing the line from which ft and 6' are reckoned to 

pass through the center of the attracting mass, we shall have for a mass 

of uniform thickness li, all integrations in (10) independent. For that 

part of the disturbed surface which lies above the undisturbed surface 

r'=r0+v will fall between the extreme values of r, which are r0 and 

r0+7tj and hence r^r' for values between r0 and TU+V, r>r' for values 

between.r0+v and r0 +7fr. For that part of the disturbed surface lying 

below the undisturbed surface r>r'. In both cases the limits of 6 are 

0, and the angular radius fi of the attracting mass and the limits of A 

are 0 and In. Therefore for that part of the disturbed surface lying 

above the undisturbed surface equation (10) gives

/

1.3 

sin m I dX

+p / sin Odd / fa

Likewise for that part of the disturbed surface which lies below the 

undisturbed surface, equation (10) gives

(115)
32 FORM AND POSITION OF THE SEA LEVEL. [BULL. 48.

For brevity, let the integrals with respect to r in (38) be denoted as 

follows :

T- _( 3 3

° 3(r0 +r) '

^ '

(40)

7- _

2~

0

4

4/0 ; /- 2 ' _(r0+/o]t-o

«7i'=(r0 +/0 (7i-»),

(41)

The ambiguous form which J"/ assumes when t=2 receives its proper 

interpretation in the third of (41). 

Similarly, let the integrals with respect to r in (39) be denoted thus:

(42)

Substituting these equivalents of (40), (41), and (42) in (38) and (39) 

the latter become, respectively,

V=p amede [(J0+J0')Po+(Ji+J1 ')^i4-. . .]dl, (43) 

Jo

F=p /'^si 

Jo , 0

(116)
WOOIWAKU.] HARMONIC DEVELOPMENT OF POTENTIAL. . 33

Now, it is known from the theory of spherical harmonics1 that the 

general value of Pt is

Pi=fi (c,os 6)fi (cos 0')+ terms multiplied by cosines of multiples of (A. A'). 

Since

,,

/

J

cos t(A A')=:

o

we have to deal only with the first term of Pt . The function of cos 0 

or cos 6' involved in this first term is defined as follows:2

2£ . 1 . 2 . . * ' d/A*

The following important relation exists between any three consecutive values of /, (/*), viz: 3

The remaining integrals in (43) and (44) are therefore of the form

J

*0 '2ir ^,1 

/ (cos 6) sin 0d0 I dM.=2j7/,(X) I ft (^)d^. 

0 «./ 0 J cos 0

Let

(46)

cos/3

The known value of this integral is4

The values of Fin (43) and (44) may now be written thus, replacing 

6' by #, #' or a being the angular distance of the attracted point from

the center of the attracting mass:

(48) 

for points of disturbed surface above undisturbed.

i-O

for points of disturbed surface below undisturbed.

1 See Heiue, nauclbuc.li tier Kugelfuuctionen, Tkeorie und Amvendungen, Ereter 

Band. Second edition, G. Reiraer, Berlin, 1878.

2 Heine, p. 19. 3 Heine, p. 91. 4 Heine, p. 93,

Bull. 48 3 (117)
34 FORM AND POSITION OF THE SEA LEVEL. [BULL. 48.

17. The values of V in the last two equations are exact, but their applicability to the problems we have to consider is limited by the slow 

convergence of the series in the second members. In these problems 

r<>, fo+'Wj and ro+h are nearly equal, and hence the convergence of the 

series depends almost wholly on the convergence of the functions of a 

and /?.

If we neglect terms of the order h/r0 , v/r0 , and upwards, equations 

(48) and (49) become identical and equivalent to (20) and (21). To show 

these facts we expand Jt and J- of (40) and (41) and obtain

t+£)! 1 + 02 ], (50) 

in which 02 represents terms of the second order. Likewise, (42) gives

o

(51)

Hence, to terms of the first order we have 1

and (48) and (49) become

t=00

V=2r0 1ip7r \ / (cos a)-F<(/?). (52)

 t = 0

Now, if cos ^=cos 6 cos a+sin 6 sin a cos (A A/),

J7r 7 /,(cosa).F<(/?)=

^Lj Jo Jo \a-2cos

But this integral, as shown by the transformation in section 3, is equivalent to

. . /cosp cos/J\ " /"Vcosj? 4 / ^^^ _ .. J dp or 4 7 '

_ cos aj * f n \cosj5-cos or

according as or is less or greater than

'This inference from (50) and (51) does not appear to be quite satisfactory. For 

large values of i, <7i+Ji' and Ji" are less than r0/i; they are each 0 for i=oo. The influence of the too great factor r0h in the higher terms of (52) is, however, counteracted by the s m all factors/<( cos a) Fi(/3) in those terms; and that the order of approximation secured in (52) is sufficient is evident from the equivalence of (52) with (20) 

and (21), -whose order of approximation has been investigated in Article IV,

(118)
WOODWARD.] EFFECT OF EEARRANGED WATER. 35

This establishes the equivalence of (52) with (20) and (21), and furnishes the following development in polar harmonics of the elliptic integrals in (22) to (25) :

(53)

z 2 =

COS » COS

 

 cos a

=

7T\ A 

=r? 7

2/ i

In order to show the form of the second numbers of equations (52) 

and (53) we give below the first four values of /(cos a) and Ft (/?), respectively. The series of values may be easily extended by means of 

the relations in (45) and (47).

=I-co$3=2 sin2

(54)

/o(cos a)=l, j

&

/i(cos ar)=cos a} Fi(/3)=^(}. cos2 /?)=£sin2 /?,

/2 (cos a)=£(3 cos2 a 1), F2 (ft)=%(e,Q& ft cos3 /?)=£sm2 /

/3 (COS a) =4(5 cos3 <*-3 cos a); #,(/?) =|(6 cos2 /5-5 cos4 ft I).

By means of these values equation (52) may be written thus:

+2 sin2 ^+icos a sin2 ft 2 2

cos2 a 1) sin2 /? cos ft

) cos3 or 3 cos a) (6 cos2 /? 5 cos4 /? !)

VI. EFFECT OF EEAEEANGED FEEE WATEE.

18. In case the disturbing mass is as large as the supposed ice mass 

of the glacial epoch, the attraction of the rearranged free water on 

itself may be appreciable. To determine the exact effect of this attraction would be a work of great difficulty even if we had the requisite 

information, namely, an accurate knowledge of the complicated shapes 

of the continents and sea bottom. But we may determine an effect 

which will exceed the probable actual effect by supposing the whole 

surface of the earth covered with a film of water free to assume the

(119)
36 FOKM AND POSITION OF THE SEA LEVEL. [BULL. 48.

proper form for equilibrium under the given forces. To fit this ideal 

case formula '(6) may be modified in the following manner:

Let the potential V in (6) be replaced by F+JF, where AY is the 

potential due to the rearrangement of the water. Likewise replace 

the constant' F0 byF0+^F0 . Then, if AV denote the corresponding 

change in v, equation (6) gives

(55)

Now, v-\-4v may be expressed by a series of Laplace's functions 

(which are in this case polar harmonics) thus:

in which c0 is a constant of small numerical value ; and hence, denoting 

the density of sea water by pw , we have, as shown by Laplace,1

V has already been expressed, equation (54), in a series of Laplace's' 

functions. Denoting these functions in (54) for brevity by Y0 , Ji, T"2 ? 

etc.,

Substituting these values of F, JF, and v+Av in (55) there results, if 

we make the obviously permissible substitution

2 p

the following equation :

-n =0.

According to the theory of Laplace's functions we must have in this 

equation the sums of the functions of the same order separately equal

1 M6canique Celeste, Book III, Chap. II,

(120)
W66bWAiiD.] EVALUATION OF CONSTANTS F0 AND U0. $7

I

»

to zero, which amounts to placing each line in the equation equal to 

zero. Thus we find

 1 _

i _ 

whence by summation

(56)

This equation expresses the total effect of the disturbing mass in 

altering the sea level, v being the effect which would result if the ocean 

were an infinitely rare fluid, and AV being the increase over v which 

would result under the assumed conditions. Obviously, v and Av may 

be expressed separately. Thus

(57)

VII. EVALUATION OF CONSTANTS F0 AND U0 .

19. We proceed now to determine the constants F0 of equation (6) 

and U0 of equations (56) and (57).

It has already been stated that these constants are to be determined 

from the condition of equality in volumes contained by the disturbed 

and undisturbed surfaces, a condition whose analytical statement is, to 

terms of the order we neglect,

0V / v sin ada Q. 

(121)
38 FOKM AND POSITION OF T.HE SEA LEVEL. [BULL. 48.

Substituting the value of v from equation (6) in this, there results

n, /->, 

/ Vsmada Vo I sin ada Q. 

t/o t/°

whence

Fo-1 / V&inada. (58)

(a) The easiest way to evaluate this integral is to substitute for Fits 

value given by equation (54). We get, then, at once

since by the theory of spherical harmonics all terms of the series except 

the first vanish in the integration. For the same reason, if we apply 

the condition

/

sn aa=

to equation (56), it will appear that

Uo=r0=2siu2 |. (60)

(b) The value of F0 may also be found by the following process, which 

is chiefly interesting on account of its complication as compared with 

the process used above. For points within the perimeter of the attracting mass replace Fin (58) by FI of (20), and for points outside the perimeter replace F by F2 of (21). Making these substitutions, there results

0=2r -2ro¥h ^y^ ( fl ismara+y^ sin ada f*I ,8in«*J. in ada} (61)

Substituting the value of Ji from (24),
WOODVARD.J EVALUATION OF CONSTANTS V0 AND U*.

Then the last integral becomes

dyi I 2t2dt

sin2 yi (t2 -!)2

in which

39

and

sin cos 71

Substituting these limits in the non-logarithmic part of the integral, 

it becomes

Integrating by parts all terms of tbis expression except the first, we get

(l-sin2 4 sii

2i

4 sin2

sin ~ sin sin

+ arc sin '( sin ^- sin yi j

 cot2 -

This gives

it/o

cot2 -g arc sin (sin ' sin y\ )

^J V ** X

sin ordfa=2(sin /? /? cos /?). 

(123)

(62)
40 FORM AND POSITION OF THE SEA LEVEL. [BULL. 48.

The second integral in (61) becomes by substitution of the value of 

72 from (25)

jr sin ada

f- I cosz

2 10

- sin2 . sin2 1-sin2 sin2

sin cos

1-sin2 sin2

sin L sin2 ;/2 cos y2

2i

1-sin2 £ sin2 y2N

yz+- sin 272 2 arc sin ( sin £ sin

^ X

( 1 sin2 ^ sin2 72 ) sin^2 

V, ^ y___

arc sin ( sin ' sin

J2 sin ada=

This gives

The sum of (62) and (63) is

sin2

(63)

which, substituted in (61), gives for V0 the same value as (59).

VIII. EQUATIONS OF DISTURBED SURFACE.

20. -By reference now to equations (3), (6), (20) to (23), and (59) we 

find for the equation of the disturbed surface when the .effect of the rearranged water is neglected

(64)

(124)
 WOODWAIUX] EQUATIONS OF DISTURBED SURFACE. 41

The corresponding expression in polar harmonics is [see equations 

(56), (57), and (60)]

(65)

Under the assumption that the water covers the whole sphere and is 

free to adjust itself as stated in Article VI, the equation to the disturbed surface is

a)Ft (/S)

i=a>

(66)

.2 y<

21. The position of any point of the disturbed surface is thus defined 

by the co-ordinates v and «, v being the elevation or depression of the 

point relative to the undisturbed spherical surface and a the angular 

distance of the point from the axis of the disturbing mass. Jin (64) 

and (66) is to be computed from (22) or (23) or their equivalents (24) 

and (25), according as the point is within or without the perimeter of 

the disturbing mass. The functions ft (cos a) and Ft (/3) are given by 

(45) and (47), respectively.

22. The general character of the disturbed surface when the effect of 

the rearranged water is neglected is evident from (64). It is symmetrical with respect to the axis of the attracting mass. It lies without or 

within the spherical surface of reference according as I is greater or

less than n sin2 £.. The values of I for the point of the disturbed sur2

face at the center of the attracting mass, for points along the border of 

the mass, and for the point 180° from the center of the mass, are given 

by equations (26), (28), and (30), respectively. If we denote the corresponding values of v by the suffixes 1, 2, 3, we get

» £( "> £-" §>

(67)

sin2 

4=

a=7t. 

(125)
42 FORM AND POSITION OF THE SEA LEVEL. [BULL. 48.

The meaning of these equations may be most readily understood by 

reference to Fig. 3. Thus, if the circle FCH represent (in crosssection) the undisturbed sea level surface of the earth, and a stratum of 

matter, as an ice cap, ABDF, be added thereto, the new sea-level 

surface will assume the form indicated by the dotted line. The values 

of Vi and t'2? as shown in the diagram, are positive, while the value of 

v3 is negative. If, on the other hand, we suppose the space A'B'D'F 

to be occupied by matter of less density than the average density of 

the earth's crust, as is the case in a lake basin, the disturbed surface 

will fall within the undisturbed surface from F to some line PQ, and 

outside the undisturbed surface from PQ to 17, i. e., Vi and v2 will be 

negative and v3 positive, or, what amounts to the same thing, p in (67) 

will be essentially negative.

23. It is of interest to inquire what angular extent of mass will produce numerical maxima of ^i, ^2 > and v3 , supposing the thickness Iti and 

the densities p and pm constant. By means of the usual criteria it is 

readily found that

maximum for /?=60°,

maximum for sin ft -, or /?=39° 32',

71

maximum for /?=120°.

(68)

24. A glance at equation (66) suffices to show that the effect of the 

free water, if it covers the whole earth, is simply to produce an exaggeration of the type of surface defined by (64) and (65). The series in

(126)
WoobwAitD.] EVALUATION OF INTEGRALS /, AND I2 . 43

the third member of (66) expressing this exaggeration is rapidly converging on account of the diminishing factor

H!s -3

Pw

which is, since ^ is about -^-,

22i+5'

The essential features of the disturbed surface are, therefore, in any 

case, defined by (64) or its equivalent (65); and in most cases the effect 

of the rearranged water may be neglected as unimportant, or as of no 

greater magnitude than the uncertainties inherent in the data for actual 

problems.

IX. EVALUATION OF THE DEFINITE INTEGRALS Ii AND I2 .

25. The equations (67) define the position of the disturbed surface in 

some of its most characteristic points. To define its position at any 

other point we must evaluate the elliptic integral I\ or J2 , which pertains to such point. These integrals have already been expressed 

[equation (53)] in a series of polar harmonics, which, if more convergent, 

would suffice for computing 7t or I2 . It is easy, however, to .derive 

more convergent and convenient series than that of (53), and this is 

the object of the present Article.

First take Ii of (24). For brevity put

and

. a

w= -.,

&= sin .

Zt

Then by Maclaurin's series, of by the binomial theorem, we readily find

7T

/»2 .' -

J1==2& / (l-Asin2 xi--Bsin4 7i-Cf sinG 71 - . . .)dyi, (69) 

. c/o

in which ,

(127)
44 FORM AND POSITION OF THE SEA LEVEL. [BULL.43.

The even powers of sin 7/1 may each be expanded in a series of the 

form

c+d cos 2yi+e cos 4/i+ . . .

in which c, <Z, e, etc., are constants. But since

cos

we shall need in these expansions only the values of c. The value of c 

in the expansion of (sin y)Zn is

_2ro(2n-l)(2n 2) . . > . (w+1) (\ 

C~ 1 . 2 . 3 . . . . n

G)2"

Applying this formula, and making the integration in (69), there results

Hence if we put1

_&2) (21+35&2+5064 +7066+105&8 +231610 ).

. a Sm 2

sm

(70)

This series converges rapidly, except for values of w near unity. In 

a practical application, to be considered presently, wherein /?=3S° (70) 

gives, using terms up to that in wu, inclusive, /i too great by about 5 

per cent, for the case w=l. But this is the most unfavorable case,

The general value of g is

2n(2w-l) (2n-2) . . . (m+1) gn _ . 

. 3* .

+1.1.3.5 . . .

+1.1.3.5 . . .

+1.1.3.5 . . .

+1.1.3.5 . . . (2n-9> .1.8. 5^=^A«

1 J * t>

+ . . . 

-1.3.5

(128)
WOODWARD.] EVALUATION OF INTEGRALS /, AND 7a . 45

and one moreover for which the exact value of Ji is known from equation (28).

26. By a process entirely similar to that followed above, the expansion of (25) gives, writing for brevity, 

^=cosec^,

If in this expression we put

5 32768~I~2T2'

we find

#6 262144 I

J2=

Sn

w= &=sin|,

(71)

This series converges somewhat more rapidly than (70). For the 

case in which /?=3S° and for the extreme value «0=1, using terms to 

that in w~u inclusive, (71) gives J2 too small by about 3 per cent.

*A general expression for the nth term within the brackets, beginning with the 

third terra, for -which n=2] is the following:

2n(2n l)(2n 2) . . («+2)

,'+1.3 .5.,. . (2n-l) 

+1.3.5. . . (2n 3) 1 . nv*

+1.3.5. . . (2n-5) 1.3.' 

+1.3.5. . . "(2n 7) 1.3.

+1.3.5. (2w 1 

(129)
46 FORM AND POSITION OF THE SEA LEVEL. [HULL. 48.

27. For points near the border of the disturbing mass I2 may be 

expressed by a more rapidly converging series than (71). Thus from 

equation (23)

cos p cos a 

uv 

Let

cos j3 cos a=2 sin ( ^ ) s^n

Then

a2

I«=1 2 =

'0

~ - -i

!____<? ______ ? _______ 8 ____ . . . L.

I (cos p cos a) (cos p cos cf)2 ( cos j> - cos a)3 I

Now, if

/v ^ _ 1 j

Jo cos 2> cos a: sin a: e gin

^=-Sin« /* 

aa A t/o

___ 

(cos # cos «)2 '

> cos a)2 ' ~^"~ " /n (cosj> cos«)3 '

etc.; 

whence X"1O 'dp

(cosj) cos a)2 siii a

/ (cosjp cos «)3 2 sin2 a

etc.

The integrals in the third and higher terms of the above series are 

thus seen to depend on the integral in the second term. Making the 

requisite differentiations we find, to terms of the third order inclusive,

. sin

= cos "a3 3-2 sin2 q 2 /9_ 4. j_ ^ 2 ^ 3 5 ' '

(72)

_/5a sin /? 3a4 cos 0- sin /? t \ 

Vlti sin2 a"1 8 siii4 a " ' V'

(130)
WOODWARD.] SLOPE OF DISTUKBED SURFACE. 47

X. SLOPE OF DISTURBED SURFACE.

28. Having derived the requisite formulas for computing the position 

of auy point of the disturbed surface, it remains to determine the slope 

of this surface relative to the undisturbed surface.

Differentiating equation (64) with respect to a, and dividing the result by the radius of the undisturbed surface r0 , we get

dv _ 3hp dl (73)

This expresses the slope or inclination of the disturbed to the undisturbed surface in a meridian plane through the center of the disturbing 

mass; it also expresses the deflection of the plumb line in the same 

plane.,

In order to apply (73) it is essential to have the general value of

dl da'

Since

and hence (70) gives

da

=-* cos

Similarly (71) gives

dI2 a

-~= n cos - da y

(74)

(75)

29. Equations (74) and (75) will suffice for the computation ofdl/da, 

except for points near to or at the border of the attracting mass. As 

a approaches equality to ft the above series become less and less convergent, and finally divergent when a=/3 or w=l. This may be most

(131)
48 FORM AND POSITION OF THE SEA LEVEL. [BULL.48.

readily seen by differentiating (22) or (23) with respect to a-, and then 

making a=fi. Thus we find

"=-i.8in<»

n.nr '2

-

da cos ft cos p 2

loge

. p sm ! 

Likewise the integrals (24) and (25) become, after differentiating th'ein 

with respect to a and then making <*=/?,

sec2 YI tan2 y\dyi

c2 ^

2

(A)

(B)

sec2 ' sec2 y2dy2

sec2 j/z tan2

fsec2 £ +tan2 

\ .^

This shows the equality of (A) and (B) since (B) is plainly infinite, 

its value being

tan

sec^

z

30. This failure of equations (74) and (75) for points at the border of 

the attracting mass arises from the fact that the expressions (20) and 

(21), though very approximate for the magnitude of the potential F, 

are not sufficiently general to give an accurate value of dV/da, or the 

attraction in the direction of the arc a for those points. To determine 

the slope of the disturbed surface at the immediate border of the disturbing mass a special investigation is requisite.

Since by equations (3) and (6) the slope is expressed by

dv dV (76)

(132)
WOODWARD.] SLOPE OF "DISTURBED SURFACE. 49

we may derive an expression for the attraction d V/r0 da directly. The 

exact expression for the horizontal attraction towards the axis of the 

mass of any element mass is, using the same notation as in Article III,

4r3dr sin2 ^ cos2 ^- dd cos Xdh 

-p 

and the integral of this is d V/r0 da. 

Now, as heretofore, let

r=r0 +w, . r'=r0+v. 

In addition put

q=.r r', 

so that

ij= v for r=r0)

Also let

0

whence

af /j>*«^ * f*ilQ e ,1// 

VI Cj -^ / Q V/vO Wi

2

Making these substitutions and neglecting terms of the order

ti vi S £ \ 2

± , 1, and ( ^-) ,

the above expression becomes

« FSazarf cos P 

Integrating with respect to 77, and substituting the limits given above, 

there results

J

cos 

Bull. 48 4 (133)

*ae i

w^t J
50 FORM*AND -POSITION OF THE SEA-LEVEL. [BULL.48.

If, now, we suppose the attracted point on the border of the attracting mass, the limits of ^ will be 0","and, with sufficient approximation, 

2r0 sin (3 cos A=c cos A, say. , Integrating with respect to £, and substi- , 

tuting these .limits, we "get

A

n ,, . . , c2 cos?'A . G cos A\ 

cos AdA Ioge (^/l^- -t + - J- ...

It remains to integrate these last expressions with respect to A between

the limits 0 and ~. An application of the formula* for integration by

2i . '

parts will readily transform them to elliptics, but since their element 

functions decrease very rapidly from the lower to the upper limit, the 

following process 1 will suffice. Consider the integral

/ cos \dk loge ( J 1+

/ m -^!:

in which A-is such that ( - ) may be neglected in comparison with \ccosAy

/ ^ N 2

unity. In the cases we have to consider [ -) will not exceed TBTT \c cos Ay

if cos A=T Ly- or A=89°25' about. Then, since ,

- c2 cos3 A ccosA

, r 

= loge |

the above integral becomes

/»\ 2c cos A 2c C^ r^- I cos A<ZA loge : = loge I cos A^A + I cos A^A loge cos A

Jo . v. , v Jo Jo

/ 2c \ 

= ( loge 1 j sin A + loge (1+ sin A) 4- (sin A 1) loge cos A".

But since sin A is very nearly unity, the last expression reduces to

The error of this integral arising from the use of A instead of ^ as the &

upper limit is less than

, c2 cos __ 2 A . c cos A \

-_-

which, if cos A=T^ and c cos A/#= LO, amounts to about ^-

1 Given in a somewhat different form by Helmert in Theorieen dec hohereu Geo.- 

dasie, Vol. II, p. 322.

(134)
WOODWAKD.] DISTURBED CENTER OF GRAVITY. 51

For the entire attraction, therefore, of the mass for a point on its 

border we get »

Finally, restoring in this last expression the value of c, viz, e= 

2r0 sin /?, (76) becomes

<">

XI. DISTURBED CENTER OF GRAVITY OF EARTH.

31. Thus far the disturbed surface has been referred to a spherical 

surface concentric with the earth's center of gravity before the disturbance arose. In determining the effects of the ice mass in glacial 

times this is the proper surface of reference, since we wish to know the 

distortion of the sea level in those times relative to the sea level in 

preceding and following epochs. If, however, it is desired to consider 

the joint effect in distorting the sea level of existing masses, like the 

continents, on the hypothesis that such masses rest on the surface of a 

centrobaric sphere, a .better surface of reference will obviously be the 

disturbed or existing center of gravity of the earth. The use of the 

latter center will require a slight modification of the preceding formulas 

defining the disturbed sea surface.

To determine the radial displacement of the earth's center of gravity 

due to the addition of such a superficial mass as we have considered, 

it is only necessary to equate the statical moment of that mass to the 

statical moment of the earth's mass, the moment plane being perpendicular to the axis of the disturbing mass at the undisturbed center of 

gravity of the earth. The moment" of an elementary ring of angular 

radius /3', measured from the axis of the disturbing mass, is to our 

order of approximation

2r0*hp7r sin ft1 cos fi'dfi'.

Hence, if ff denote the displacement sought and M the earth's mass,

f*0 

MG=r<?lip7t / 2 sin 8' cos 8'A8'

J°

=r(?hp7t sin2 ft.

, Therefore, by substitution of the value of M given in equation (3), we 

find

<T=|A-£- sin2 8. (78) 

(135)
52 FORM AND 'POSITION OF THE SEA LEVEL. [BULL. 48.

Now, the elevation of any point of the disturbed surface relative to. 

the sphere in the'new position will be less than its elevation relative to 

the sphere in the former position by an amount whose value to the 

proper degree of approximation is

ff COS Of,

a being, as heretofore, the angular distance of the'point from the axis 

of the disturbing mass. That is, if v' denote what v becomes by the 

change in position at the sphere of reference,

V' = V ff COStf.

Hence, by virtue of (64) and (78) we find for the equation -of the disturbed surface when the sphere of reference is concentric with the disturbed center of gravity of the earth,

(79) v

And the inclination of the disturbed surface to the surface of reference is

_?L =3fe P. I-T-+ - sin a sin2 fi\ (SO) 

v da 4 ' ) ^ '

XII. EQUATIONS OP DISTURBED SURFACE WHEN THE DISTURBING 

MASS IS OF VARIABLE THICKNESS.

32. Throughout the preceding investigations the thickness of the 

attracting mass has been considered uniform. On this account the 

range of application of the formulas derived .is somewhat narrow. It 

may be remarked, however, before proceeding to extend the investigation to more complex masses, that inasmuch as the data for actual 

problems will be in general more or less uncertain, or to a large extent 

ideal, formulas of a more comprehensive and hence more complex character are not specially desirable. Approximate calculations of a rather 

rough sort in some cases' will be as good as the data for those calculations. The effects assigned by the foregoing equations will be for the 

most part in excess of the probable actual effects, and in so far as computation can contribute arguments pertinent to observed facts the maximum effects will be most essential. On the other hand it will be 

desirable in some cases to get an idea of the inferior limiting effects. 

The most important of these cases relates to the extent of submergence 

attributable to the ice cap of the glacial epoch. There would seem to 

be little probability of uniform thickness in such a cap. Apparently

(136)
WOODWARD.] EFFECT OF MASS OF VARIABLE THICKNESS. 53

some sort of regular decrease in thickness (with here and there considerable though comparatively unimportant deviations) from the center 

to the perimeter of the mass is more probable. Such a law of decrease 

is expressed by the equation

<p(/3)=li=li0 (81)

in which h is the thickness along any radial line whose angular distance 

from the axis of the mass is /?, JiQ is the thickness along the axis, and /30 

is the angular radius of the perimeter of the mass. In brief, li is a function of /?, as stated by the first member of the equation. The exponent 

n must be a positive number and may be for our purposes restricted to 

integer values. In order to determine the effects of masses conforming to the above, and, in general, any law requiring symmetry of mass 

with respect to a radial axis, we shall devote the present Article to the 

necessary extension of the formulas already derived. 

33. The differential of equation (64) with respect to ft gives

This expresses the elevation of the disturbed surface due to an annu. 

lus of angular radius/?, of angular width <?/?, and height ft, the density p 

* being uniform. If in this equation-we make li a" function of /?, or write 

h=g}(/3), and integrate between the proper limits, the result will be the 

elevation of the disturbed surface due to a mass whose thickness conforms to the law expressed by <p(/3}. Calling, for the sake of distinction, v" the new value of the elevation of the disturbed surface, and 

the proper limits of /?, /?1} and /?2 , the result of this integration is

*»=*&.

Tins equation assigns the effect of any homogeneous mass whose 

bounding surface is one of revolution about a radial axis, subject to 

the restriction that the maximum thickness of the mass may be neg-. 

lected -in comparison with the radius .of the earth. It is obvious, 

however, that the .integral in (83) may be impracticably complex for 

some forms of <p(fl}. To avoid undue complexity andcat the same time 

attain results suitable for our special purposes we shall here confine

(137)
54 FORM AND POSITION OF THE SEA LEVEL. [BULL. 48.

attention to that form of q>(/3) expressed by equation (81). For this 

function we have, considering the whole mass,

#=0, #=/70 , 

=0 for?=,'

sin

Then, observing that J=0 for 6=0, (83) becomes

7tpn

The definite integral in this expression depends on and will in general be no less complex than J, -which is defined by (22) to (25). An 

examination of (24) and (25) shows that for points of the disturbed 

surface within the perimeter of the disturbing mass

ft/0

J(sin£) dsin£ = / I2 ( sin'

' V ^

(85) 

V

"For points of the disturbed surface without the perimeter of the 

mass it is only necessary to replace I in (84) by /2 of (25), or replace 

the limit a in (85) by /?0 . By means of the series (70) and (71), or the 

harmonic series (53), (84) may be evaluated for any point of the disturbed surface.

For two points of the disturbed surface, namely, at the center of the 

mass and 180° from that center, (84) yields to direct integration. The 

process of evaluation is as follows :

The integral in (84) is a function of «, /?0 , and n. Let it be symbolized 

\)yf(a/30n). For the two points noted above let this function be distinguished by the suffixes 1 and 2, respectively, so that it becomes

/i(or, /?o, n) for <?=0,

and

/z (a, /?o, n) for a=7t.

Then, since from £26), 1= n sin ^ for «=0,

8in?. (86)

(138)
WOODWABD.I EFFECT OF MASS OF VARIABLE THICKNESS. 55 

Likewise, since from (30), I==2/r sin2 L for a=7t,

/

ox /) _ « 

/8in|\ , 

( & 1 d8in J' * 

\ sin £2 / 

V 2/ 

7T/^ /5 /? A

=T( 4 2 cos '-~f fa cosec ^-H ) for w=l,

=~ 4 cos8 ° -lan2 > for w=2, 4 4

When w is more than a few units the integral in (87) may be evaluated by rapidly converging series. Thus, call the required integral J5, 

and let

sin = x sin °

Then

B= \ £ |a«in'4f

0

1

n+1

(88)

34. It will be particularly essential for our purposes to evaluate (84) 

'for points outside the border of the disturbing mass. The integral required is, if we write for brevity

w&n n I IJ>n db. 

Jo

i Now, J2 from the equation preceding (71), may be written thus:

(139)
FORM AND POSITION OF THE SEAv LEVEL. (BULL. 48.

in which ji,j2i etc., are obvious functions of a or y=cosee a. Therefore 

the above integral becomes

nb -w rbo ri n-l,,

, I I2 b db=n 

J o

w+2 1 

%

w

(89)

+ 

= 7tS, say.

35. Let the value of 8 in the last equation for points at the border of 

the disturbing mass where a=/?0 be denoted by Si . Also denote by 

 fli", i?/', and %" the elevations of the disturbed surface at the center of 

the mass, at its border, and 180° from its center. Then equation (84), by 

means of the results in (86) to (89), gives the following equations analogous to the group (67):

sin

pm \n+l n+2

(90)

-q 2 sin2 &-'. 

PA 4

sin2 ^

36. To define the slope of the disturbed surface it is in general necessary to differentiate (84) with respect to a. The result is of a complex character and subject tp discontinuity for points at the border of 

the mass. For practical purposes, however, it will suffice to make use 

of Av"/da instead of the differential coefficient, and thus determine 

average slopes over some finite portion of a meridian section of the 

disturbed surface.

37. In discussing the disturbance of the sea level attributable to the 

ice mass of the glacial epoch, it will be of interest to estimate the effect 

of the rearranged free water. For this purpose we may extend the 

second term of the third member of equation (66) so as to make it assign the effect of the rearranged water when the mass is of variable 

as well as uniform thickness. The process is strictly analogous to that

(140)
EFFECT OF 'MASS OF VARIABLE THICKNESS. 57

followed in deriving (83) and (84) from (64). Thus, confining attention 

to the form of mass defined by (81), and writing, as in section 34,

1 

and

we readily find from (60)

rbo a)nbo~n I F^b"- 1

\J 0

db

(91)

^ few values of Ft (b) derived from (45) and (47) are the following: 

=2(b2 - &4 ),

2&6 ),

- 56"),

- 35&8+ 14&10),

From these the corresponding integrals

/

bo « 1 

Ff (b)b db

can be readily derived. Thus, for example,

F5 (b)bn~l db=2n,

-K-^&o , 70 7 r6

126 , 

'w+10

(141)

(92)
58 FORM AND POSITION OF THE SEA LEVEL.

38. A common property of the formulas (84) to (92), both inclusive, 

is worthy of notice. They all refer to a mass whose thickness conforms 

to the law (81), namely,

When w=co this gives (p(/3)=h=h0 , or the thickness of the mass is 

uniform. Therefore the formulas (84) to (92) should return to the forms 

applicable to a mass of uniform thickness on making n=x . Such is 

the case. Thus (84) becomes (64), as is readily seen by an application 

of the formula for integration by parts. Likewise, for n cc .equations 

(86), (87), and (89) become (26), (30), and (28), respectively, and the group 

(90) assumes the simpler forms of 'the group (67).

B. APPLICATIONS.

XIII. RELATIVE POSITIONS OF LEVEL OE EQUIPOTENTIAL SURFACES

IN A LAKE BASIN.

39. Consider the question stated in section 23 (a), relative to the level 

surfaces in a lake basin. In this case it is required to determine the difference in elevation at the center of the basin of two level surfaces 

which intersect along its perimeter. The first two of equations (90) 

give

< 93>

This. represents the difference in elevation of a level or liquid surface 

at the center and at the border of the basin.,, p must be understood as

rifc

the excess or defect in density of the liquid relative to the average 

density of the superficial strata of the earth. Thus, if the liquid in 

question be water,

/>= (£&» -.1)= 1-8, approximately. 

If we differentiate (93), regarding p as variable, the result is

sin Q\ ^ (94)

' Pm\ n+L

This expresses the required separation of the two level surfaces in 

question ; i. e., the separation at the center of the basin of two level surfaces which intersect at the border and which are the free surfaces of 

two liquids whose difference in densitj7 is zip.

To illustrate more fully the meaning of (93) and (94) let ABDE in 

Fig. 4 represent a cross section through the axis of the basin. ACS is 

a circular arc parallel to the section of the sphere of reference. If the 

basin be filled with water the section of the water surface will lie be. 

low A CB as AG'B. If the water be removed the corresponding sec-

(142)
WOODWARD.] EQUIPOTENTIAL SURFACES IN A LAKE BASIN. 59

tion of the level surface through A and B will fall below AC'B as AC"B. 

Hence, iu the diagram

FIG. 4.

40. To get some numerical values, let us assume the followiDg data:

7to=l,000 feet, /?0 =arc of IP, /ow =5-5, 

p = 1.8, z/p=l,

this latter being approximate difference iu density of air and water. 

Then.

Pm

3/^=545 feet.

Pm

The 'assumed value of /?0 is equivalent to about 69 miles measured 

along the earth's surface. .

Now, by means of equations (86) to (89), we find the following table 

of results corresponding -to several values of w. which defines the shape 

 of the basin see equation (81). The results in the fifth column of the 

table express the difference in elevation of a water surface at the center 

and at the border of the basin; and those in the sixth column express 

the depression of the level surface at the center of the basin consequent 

upon substituting air for water as the attracting mass.

o

Table o/ values showing relative positions of level surfaces in a lake basin 140 miles in 

diameter and of 1,000 feet maximum (dxial) depth.

n

1

2'

3

4

5

8

7

8

9

10

00

_5_sin!i

71+1 2

0. 00436

582

654

698

727

748

764

776

786

793

0. 00873

Si

0. 00161

245

298

333 359'

.379

395

407

418

427

0. 00556

-^sin^-S, 

n+l 2 '

0. 00275

337

356

365

368

369-

3G9

369 '

368

3G6

. 0. 00317

va" Vi"

Feet. 

2.70

3.31

3.50

3.58

3.61

3.62

3.62

3.62

3.61

3.59

3.11

A (V'i" Vi")

Feet. 

1.50 

1.84

1.94

1.99

?. 01

2.01

2.01

2.01

2.01.

1.99

1.73

(143)
60 FORM AND POSITION OF THE SEA LEVEL. [Bm.«

It will be observed that the differences in the last two columns of the 

above table rise to a maximum between the arguments n=5 and ?&=9. 

That they should do so is evident from an inspection of equations (89), 

(93), and (94).

The deflection of the plumb line toward the land along the border of 

c uch a lake, supposing it of uniform depth or %=oo, would be by 

equation (77) 11" or 16", according as the basin is filled with water 

or air, and the slope of the water surface at the immediate border 

would be 0.28 feet per mile.

41. An interesting inference, which might be drawn from the solution 

of the above problem, is that in triangulating a large lake we should 

expect to see from shore to shore with somewhat less elevations of the 

points of observation than the usual formulas for inter visibility of 

points on the earth's surface would require. This would be a correct 

inference, however, only in case the defect in potential due to the water 

in the lake basin is not offset by an excess in potential due to some 

local or general distribution of matter within the earth's crust.

XIV. VARIATIONS OF SEA LEVEL ATTRIBUTABLE TO CONTINENTAL 

GLACIERS OR ICE CAPS.

42. As a second application of the preceding theory we shall investigate the attractive effects of the ice mass of the glacial epoch, assuming that the earth's crust did not yield to the pressure of the ice. This 

is the problem of section 2 (&).

This problem in its physical aspects presents two difficulties, the first 

of which has not been alluded to in the foregoing sections, and the 

second only partially considered.

The first of these difficulties is to account for the enormous quantity 

of water required to form such an ice mass as is supposed to have covered our northern hemisphere during the glacial period, This mass has 

been usually estimated as not less than 5,000 feet thick at its center, 

and to have extended 30° to 90° from that center. It is. generally assumed to have diminished in thickness with some approximation to 

regularity from the center to the perimeter. The superior limit for this 

shape of mass would be a sheet of uniform thickness, and the inferior 

limit a meniscus increasing slowly in thickneSvS from its perimeter 

towards its axis. Evidently, if the water forming such a mass were 

drawn from the ocean, the latter would undergo a considerable diminution in elevation, and this diminution might nearly counterbalance the 

attractive effects of the ice in elevating the water along its border. The 

view advanced by Dr. Groll, however, assumes that there is an alternation of glaciation at the poles, the epoch of minimum ice cap at the 

one corresponding to the epoch of maximum ice cap at the other. This, 

granting the sufficiency of ice in the two caps, would make the quantity 

of free water in the ocean substantially constant. In ,the absence of 

definite information ou this point, it must be admitted that considerable

(144)
WOODWARD-] EFFECTS ATTRIBUTABLE TO CONTINENTAL GLACIERS. 61

uncertainty may properly be attributed to our computed variations of 

sea level, although the equivalent lowering of the sea, if the water in 

the ice is drawn therefrom, will be determined for each assumed mass.

43. The second difficulty arises from the fact that such a large mass 

as we seem compelled to assume for the ice cap would produce an extensive rearrangement of the sea water; and we ought therefore, iu 

computing the potential at any point of the disturbed surface, to take 

account of this rearrangement. We have shown how to do so in an 

ideal case, which presents effects for the elevation or depression and 

slope of the disturbed surface greater than the probable actual effects. 

The actual effects, as we shall indicate, probably lie about midway be- ' 

tween those assigned by the formulas for the ideal case and those assigned by the formulas which neglect the potential due to the rearranged 

free water. But on account of the difficulty in fixing an exact limit for 

the actual effects, our computed results will be subject to a small range 

of uncertainty, which may be regarded, however, as no greater than 

the inherent uncertainty in the more important data of the problem,

44. Let us now take for the mean density of the earth pTO , for the 

density of ice p, for the thickness of the ice along its axis /t0 , and for 

the angular extent of the mass /?0 , the following values :

pm =5.5, ho= 10,000 feet, 

p =1, 1 /?0 =arc of 38°.

This value of /?0 corresponds to about 2,600 miles measured along the 

earth's surface ; it is also very nearly that angular extent of mass, of 

uniform thickness, which produces the maximum upheaval of water 

along its border. (See section 23. ) 2

45. As we shall compute the effects of masses corresponding to several values of the index n, equation (81), it will be of interest to define 

with some precision the shape of the exterior surface of each mass. To

do this it will suffice to give the slope of the surface of any mass at

 e-^ 

1 This value for the density of ice is about 8 per cent, too great; but by using it 

in the formulas which do not take account of the potential due to the rearranged 

free water, we shall get results differing only slightly from the probable results.

2 If the water in the ice cap were drawn wholly from the free sea water the angular 

radius of a mass producing maximum upheaval along its border would be much less 

than the value given by the second of equation (68), for in this case the rise in sea 

level due to the attraction of the cap would be offset partly by the fall due to the 

withdrawal of the water. The difference between the rise and fall just mentioned is 

V2" of (90), minus r of (96), or

- -' 8iQ2 °-

POT \ w + z 2 / n + z 2

The value of @o which will render this difference a maximum is easily found, but 

the result is of little interest since the corresponding mass would produce effects 

much smaller than the possible effects of continental glaciers.

(145)
62 FORM AND POSITION OP THE SEA LEVEL. IBULL. 48.

several points in a meridian plane. The differential of equation (81) 

gives

dli ___nhoi ~ ."2 (95)

This expresses the slope or inclination in a meridian plane of the 

bounding surface of the attracting mass to the spherical surface of reference. Using the above values of h0 and /?0 , and for the radius of the 

'earth (see section 4) log r0=7.320120, the following table of values has 

been computed. The slopes are expressed in feet per mile. Several of 

the curves -whose slopes are given are delineated (with greatly exaggerated radial scale) in Fig. 5,

Table showing meridian slopes of exterior bounding surfaces of assumed ice masses.

equations (81) and (95).]

[See

n

1

2

3

4

5

6

7

8

9

10

oo

Slopes in feet per mile corresponding to /3 =

0°

Feet. 

3.88

0.00

.00

.00

.00

.00 

.00 *

.00

.00

.00

0.00

5?

Feet. 

3.88

1.04

.21

.03

.01

.00 it 

.00

.00

.00

.00

0.00

10°

Feet. 

3.86

2.07

.83

.29

.10

.03 

.01

.00

.00

.00

0.00

15°

Feet. 

3.85

3.08

1.85

.99

.49

.24 

.11

.05

.02

.01

0.00

20°

Feet. 

3.82

4.07

3.26

2.32

1.55

.99 

.62

.47

.23

.13

0.00

25°

Feet. 

3.78

5.04

5.02

4.45

3.70

2.95 

2.29

1.73

1.30

-.96

0.00

30°

Feet. 

3.75

5.96

7.10

7.53

7.48

7.14 

6.62

6.02

5.38

4.75

0.00

35°

Feet. 

3.70

6.84

9.47

11.66

13.46

14.92 

16. 08,

16.97

17.64.

18.10

0.00

38°

Feet. 

3.67

7.34

11.00

14.67

18.34

22.01 

25.68

29.34

33.01

36.68

00

The numbers in the above table'show that for n=l the slope of the 

bounding surface, as defined by equation (81), is steepest at the axis of 

the mass and decreases slowly from the axis towards the border. For 

values of n greater than unity the bounding surfaces slope up with 

decreasing rapidity from the border to the axis of the mass, the amount 

of slope diminishing to zero at the axis in each case. The features 

here enumerated will hold for any extent of mass, i. e., for any value of 

/?<> It will be observed also that the slope for any value of n is directly proportional to the axial thickness Ii0 . Hence the slopes corresponding to any other thickness.than that assumed (10,000 feet) may be 

readily computed from the table.

46. As to the actual form of the bounding surface of the ice mass 

of the glacial epoch we have no precise information. It is generally

(146)
WOODWAED.J EFFECTS ATTRIBUTABLE TO CONTINENTAL GLACIERS. 63

assumed, however, that,the mass was thickest along its axis, and that 

the thickness decreased with some approach to regularity between the 

axis and the border.1 The slope of ascent along the border has been 

estimated as 10 to 35 feet per mile.2 Nordeuskjold observed a rise of 

7,000 feet in 280 miles from the border of the ice plains of Greenland.3

FIG. 5.

Scale for section of sphere=s-Tn7rl0-^Tn7>

Scale for disturbing mass =TiT>W=«Ttf$feinr»

n=lio$ 1 sm"i| j, /, 0=10,000 feet. ' ( sin"£/?0 5

For curve 1, n=l; for curve 2, n=2; for curve 3, n=10\ for curve 4, w=x>.

This corresponds to an1 average slope of 25 feet per mile. It seems 

most probable that the slope of such a mass would be steepest at its 

border and diminish gradually towards its center. Whatever may have 

been the actual slopes it is thought that the preceding table affords a 

sufficiently comprehensive variety. Our equations, it is true, will assign the effects of an indefinite variety of other forms, but in the 

absence of more complete actual data the simple forms whose slopes 

have been computed are considered adequate.

'The directly opposite view is maintained by W J McGee. See his paper on 

Maximum Synchronous Glaciatiou, Proceedings of the American Association for the 

Advancement of Science, Vol. xxix, 1880.

2 Croll, Climate and Cosmology, p. 244.

3 16^., p. 245

(147)
64 FORM AND POSITION OF THE SEA LEVEL. I BULL. 48.

47. In order to form an idea of the amount of water congealed in the 

masses whose shapes have just been defined we will compute the thick, 

nesses of spherical shells of radius r0 (radius of earth's surface), having 

equal volumes with the ice masses, respectively, As the sea covers 

about three-fourths of the earth's surface the products of these thicknesses by f will represent approximately the necessary lowering of the 

sea level if the water in the ice cap~is drawn from the sea. Since the 

thickness of our assumed ice mass along any radial line at an angular 

distance /? from its axis is by equation (81)

the volume of this mass will be expressed by the integral

sin/M/?=4. sn* .

A

The thickness r of a, spherical shell of radius r0 and equal volume 

with the above is' given to a sufficient degree of approximation by the 

relation - .'

4r02 r;r=4

whence

n-{- &

sn

(96)

From this equation with our working values, /i0=10,000 feet and /?0= 

38°, we find the following values of r and f r, corresponding to the 

values of n in the preceding table:

Table showing thicknesses r of spherical shells of equal volume ivith assumed ice masses, and 

. equivalent lowering of sea level $r.

n

1

2

3

4

 5

6

7

8

9

10

00

r

Feet.

353

530

636

707

757

795

824

'848

867

883

1060

$r

Feet.

471

707 '

848

913

1009

1060

1099

1131

1156

1177

1413

(148)
WOODWARD.] EFFECTS ATTRIBUTABLE TO CONTINENTAL GLACIERS. 65

48. Although, as seen from the last table, the amount of water nee? 

essary to form such an ice cap as we are considering would, if drawn 

from the sea, cause a decided lowering thereof, yet the mass of ice 

would be very small compared with the mass of the earth. Thus, for 

example, the mass of a sheet of uniform thickness 10,000 feet and 38° 

angular radius is only 3-^00^ Part °f tne earth's mass.

49. To determine the position of the disturbed relative-to the undisturbed surface it will be sufficient to compute the elevations of the 

water at the center of the attracting mass, along its border, and 180° 

from the center, by means of formulas (90). As the slope of the disturbed surface near the border of the mass is of most importance, we 

shall compute the elevation of the disturbed surface for a circle of 

points 1° distant from the border, and thereby deduce the average 

slope of the disturbed surface within that distance (69 miles) of the 

border. The separate quantities required in this calculation are given 

in the following table for the same values of n as those used in the two 

preceding tables. It will be remembered that ?t=oo corresponds to an 

attracting mass of uniform thickness. The values of 8 in the last column of the table have all been computed from, equation (89), except the 

one for W=QO , which has been derived from (72).

Table of numerical values of functions defining position of disturbed surface.

n

1

2

3

4

5

6

7

8

9

10

oo

-ILein*! 

n+1 2

0. 16278

.21701

.24418

. 26045

.27131

. 27906

.28487

. 28939

.29301

. 29597

0. 32557

_^_ siu* *L 

ra+2 2

0. 03533

. 05300

. 06360

. 07066

. 07571

. 07950

.08244

.08480

. 08672

-.08833

0. 10600

-Si

0. 0.6078

. 09249

.11218

. 12568

. 13555

. 14310

. 14908

. 15393

' .15795

.16135

0. 21111

2(sin«£-B)

0. 01796

. 02C99

. 03243

. 03606

. 03867

. 04063

. 04216

. 04338

. 04438

. 04522

0. 0544.8

S

0. 05885

. 08916

.10842

. 12139

. 13085

. 13809

. 14380

. 14843

. 15227

. 15550

0. 19846

For brevity make the following substitutions:

81 Ua £2A

n+2

sin2 £2

Bull. 48 (149)
 6 , FORM AND .POSITION OF THE SEA LEVEL. [BULL.48.

;Then from the preceding.table we get the following table of values:

n

1

2

3

4

5

6

7

8

9

10

oo

tfi

+0. 12745

+ .10104

+ .18058

+ . 18979

+ .19560

+*". 19950

+ .20243

+ .20459

+ .20629

+ .21)764

+0.21957

**

+0. 02545

+ .03949

+ .04858

+ .05502

+ .05084

+ .06300

+ .OC664

+ .00913

+ .07123

+ .07302

+0. 10511

Ar3

-0. 01737

- .02601

- .03117

- .03460

- .03701

- .03887

- .04028

- .04142

- .04234

- .04311

-0.05152

#4

+0. 00193

+ .00303

+ .00376

+ .00429

+ .00470

+ i 00:01

+ .00528

+ .00550

+ .C0568

+ .00585

+0. 01265

Now, the factor by which we must multiply JYl7 N2 , and JV73 to get the 

elevation of the disturbed surface above the undisturbed at the center 

of the mass, along its border, and 180° from its center is

1 37^=5454.5 feet; />»

and one sixty-ninth part of this factor multiplied by JT4 will give the average slope per mile of the disturbed surface within 1° of the border of 

the ice. Hence we get in the table below the results corresponding 

to the several values of n. The plus sign indicates elevation and the 

minus sign depression of the disturbed relative to the undisturbed 

surface.

i

Table showing effects in distorting the sea level of ice caps of the same angular radius, 38°, 

andsameaxial thickness, 10,000 feet, but of varying external slopes, defined by equation (81).

i

'

Position of disturbed relative to undisturbed surface.

71

!

i

'1

'2

'3

4

5

6

i

8,

9'.

10 \

\ oo 1

At centerof 

ice mas.

Feet.

+ 605

+ 895

+ 985

+1035

+ 1067

+1088

+1104

+1116

+1125

+1133

+1198

A Ion 2 border 

of ice mass.

Feet.

+139

+215

+265

+300

+326

+347

+363

+377

+389

+398

+573

180° from center of ice 

muss*

Feet.

 95

 142

 170

 189

 202

 212

 220

 226

 231

-235

 281

Average 

slope per

mile of disturbed surface within 

1° of tbe 

border of the

ice mass.

Feet.

0.15

.24

.30

.34

.37

.40

.42

.43

.45

.46

1.00

(150)
WOODWARD.] EFFECTS ATTRIBUTABLE TO CONTINENTAL GLACIERS. 67

50. As already explained, the numbers in the second, third, and fourth 

columns of the last table assign the position, and those in the fifth column the meridian slope of the disturbed surface relative to the undisturbed surface, assuming that the ratio of the density of ice to the earth's 

mean density is j^-, and neglecting the effect of the rearranged free 

water. If ^ were the correct ratio of the densities, and if the sea covered the whole surface of the earth, and were free to arrange itself in conformity with the attractive forces, formulas (66) and (91) show that the 

numbers in the second, third, fourth, and fifth columns of the table 

should be increased by about 15,18, 26, and 18 per cent., respectively, of 

their stated amounts. But the ratio -£f is too great by about 8 per cent., 

so that for this reason the above percentages must be reduced to 7,10, 

18, and 10, respectively. Again, only three-fourths, at most, of the 

earth's surface is covered with water, so that the reduced percentages 

must be diminished to three fourths their stated amounts. Moreover, 

the sea could not penetrate the ice mass in such a manner as to produce 

the assumed increase in the potential within its border, and hence the 

reduced percentages must be still further diminished. It is estimated 

that the actual effect can not be greater than two-thirds that which 

would follow if the water were unrestricted. Accordingly, f . i^=£ tue 

above reduced percentages, or 3.5, 5, 9, and 5 per cent., respectively, 

would appear to be liberal allowances for the effect of the free water 

in exaggerating the deviation of the disturbed from the undisturbed 

surface at the points designated, over and above the tabular deviations. In view of the smallness of these possible increments to our 

computed quantities we need give the question of the effect of the rearranged free water no further consideration.

The results in the second column of the table are the heights to 

which the water would rise at the center of the ice mass if brought 

within it in any manner, as by a canal, and left free to assume equilibrium. If the amount of free water were sufficient it would rise or 

fall to the extent indicated in the third and fourth columns, respectively, 

at points along the border and at the antipodes of the center of the 

mass.

The slopes given in the fifth column will apply to isolated bodies of 

free water adjacent to the ice mass, and also to the sea surface in the 

samo vicinity, whether there be sufficient water to rise to the height 

indicated in the second column or not.1

1 Recent observations of the beaches of bodies of water contiguous with the ice 

fields of the glacial epoch, indicate uniformly that the water surfaces sloped upwards 

toward the ice. In a careful exploration of the beaches of Lake Agassiz, an extinct 

lake which lay in the valley of the Red River of the North during the glacial epoch, 

Professor Uphara has found slopes varying from zero to 1.3 feet per mile. (See Bulletin No. 39 of the U. S. Geological Survey, on The Upper Beaches and Deltas of the 

Glacial Lake Agassiz, by Warren Upham.) Near the south shore of Lake Ontario, in 

New York, Mr. G. K. Gilbert has observed slopes as great as 5 feet per mile. (See 

Science, Vol. I, p. 222.)

(151)
68 FORM AND POSITION OF THE SEA LEVEL. [BULL. 48.

The maximum possible slope of the disturbed surface would occur at 

the immediate border of a mass of uniform thickness. This slope is by 

equation (77) for /i=10,000 feet and /?=38°, 1.80 feet per mile, and corresponds to a plumb line deflection of 72". Since ifc is not probable 

that the ice cap presented at its border anything like a vertical wall 

10,000 feet high, we infer that a mass'whose maximum or axial thickness is 10,000 feet would be quite inadequate to produce a slope of 1.8 

feet per mile.

Again, since all the results in the table are proportional to the axial 

thickness of the ice, to produce as great-an average slope as 2 ieet 

per mile within 1° (69 miles) of the border of a mass having the more 

probable slope defined by the index n=Q to w=10, would require an 

axial thickness in round numbers of 50,000 feet, or 9£ miles. The 

slope at the immediate border in the extremely improbable case of a 

uniform thickness of 50,000 feet would be 5x1.8 feet=9 feet per mile.

51. To produce slopes as great as 4 or 5 feet per mile for any distance 

from its immediate border an ice cap must have great thickness, which 

implies for any large areal extent a heavy draft on the visible supply 

of water. The minimum thicknesses of ice masses of varying radial 

extent which would produce an average slope of 5 feet per mile within 

1° (69 miles) of their borders are shown in the following table. These 

values are computed on the improbable supposition that the masses 

are of uniform thickness. The volume of each mass is indicated by the 

equivalent lowering of the sea level, or %r deduced from equation (96), 

the value of n being infinite.

Table showing for ice masses of varying radial extent the minimum thicknesses requisite to 

produce an average slope of 5 feet per mile within 1° of the borders.

AngTilar radius 

of iiia^s.

o

10

20

30

38

Minimum thickuess.

Feet.

C9, 400

52, 500

52, COO

50, COO

Equivalent 

lowering of sea 

level, iiulicatiu.s volume 

of mass.

Feet.

101

2,308

4,699

7,065

For masses having moderate surface slopes,near their perimeters tne 

axial thicknesses must be about twice a,s great as the minimum values 

given in the table to produce the same average slope of 5 feet per mile 

within 1° of the borders. Thus, for the angular extent 38°, and 

for w=6, say, an average slope of 5 feet per mile would require an 

axial thickness of about 125,000 feet, or 24 miles. This corresponds to 

a lowering of the sea -of about 2.4 miles, and if the quantity of free 

water were sufficient, to an elevation of the sea along the border of th& 

mass-of about 4,000 feet.

(152)
WOODWARD.], EFFECTS ATTRIBUTABLE TO CONTINENTAL GLACIESS. 69

52. Confining further investigation to our working axial thickness, 

10,000 feet, angular radius 38°, and ratio of densities -&- , we may inquire as to the extent of the variation in sea level at any point of the 

earth's surface on the supposition of an alternation of glaciation at the 

pcles. For this purpose it is simply necessary to compute the elevation of the disturbed surface at the angular distance a of the point in 

question from the pole or axis of the ice cap and for the point ISO0 a 

and take the difference between the results. Since the maximum effects 

are of most interest, we shall compute the variations in sea level on the 

assumption "that the ice cap is of uniform thickness. The results, as 

may be inferred from the second table on p. 66, will not differ materially, except for points near the border of the ice, from the results which 

would be derived on the assumption of a sloping mass corresponding 

to the index »i=6 to 7i=10.

To compute the required integrals we may use formulas (70) and (71). 

For/?=38° we-fiiid

log #!=-- 9.3493, log/L-!=9.7049,

log #2 =8.7422, log fcj=8.8079,

log #3=8.3452, log ^=8.3849,

log #4=8.0708, log 7^=8.1186,

log #5 =7.8644, ' log fr5 =7.8904,

log #6 =7.6987. log 7rG = 7.7033.

From the formula

sin"

we compute log w for a= 10°, 20°, 30°, etc., to 180°, and then the values

7, or, more conveniently, the values ofl/Tt sin c required in (64), readily > 2

follow. The results are given in tabular form below for circles at intervals of 10° from either pole.

(153)
70 FORM AND POSITION OP THE SEA LEVEL. . [BULL.

Table showing disturbance of sea level attributable to an ice cap of 38° angular radius and 

10,000 feet uniform thickness, and variation in sea level attributable to same mass on the 

hypothesis of an alternation of glaciation at the twopoies.

Angular distance 

from either pole, 

or a.

o 

0

10

20

30

40

50

60

70

80

90

10J

110

120 ' IM

1'40

150

ico

170

ISO

Elevation of disturbed above undisturbed surface, 

or^a.

Foot. 

+1108

+ 1109

+ 1076

+ 893

° + 441

+ 185

+ 46

- 43

- 106

- 1.J2

- 186

 213

- 233

- 249 -

- 201

- 270

- 276

- 280

- 281

Variation in sea level 

from epoch of minimum to epoch of maximum glnciation, or

Va. Vigo a.

Feet. 

Ii79

14.49

1352

1103

702

 m

279

170

SO

10

80

no

279

434

702

1163

i::52

1419

1479

A graphical representation of the results in the last two columns of 

the above table is given in Fig. 6, the section of the disturbed and undisturbed surfaces being so developed that the great circle of the latter 

appears as a straight line, AB. The distances between the two curves 

representing the disturbed surfaces measured at right angles to the 

axis AB indicate the variation in sea level between the epochs of maximum and minimum glaciation at either pole.

The variation in slope of the sea surface at any point during the interval between the extremes of glaciation will .equal the sum of the 

slopes at that point for the two epochs. The maximum variation is, 

however, only slightly greater (about 1 per cent.) than the maximum 

slope already computed, and requires, therefore, no further consideration. That portion of the disturbed' surface having the greatest slope 

is made clearly apparent by the curves in Fig. G.

' ISO' 1W170' IK

VncKsturbed Surface.

Disturbed, Surface.

FIG. 6. 

Horizontal scale, jsrii&uTTtfTr > vertical scale, ^
WOODWARD.] INVESTIGATIONS OF PKATT. . 71 

XV. HISTORICAL NOTE.

53. The effect of the glacial accumulation in disturbing the sea level 

has been the subject of considerable discussion. The discussion was 

started apparently by Dr. Croll about 1866; and from the widely differing results quoted by him in his Climate and Time, Chapter XXIV, one 

might infer that the mathematicians who have attacked the problem 

are completely at loggerheads. There has been, indeed, some diversity 

of opinion as to the proper method of treating the question, %and the 

work of one writer at least is quite erroneous; but part of the apparent discrepancy in the results quoted by Dr. Croll is due also to a radical difference in the data used as a basis for calculation.

The mathematicians whose writings on this subject appear to be 

worthy of especial consideration are Archdeacon Pratt, Mr. D. D. 

Heath, and Sir William Thomson. Their investigations may be found 

in the Philosophical Magazine for 1866 j Vote: XXXI and XXXII. 

Those of Archdeacon Pratt are reproduced substantially in his Figure 

of the Earth, fourth edition, pages 236 to' 238, while those of Sir William Thomson are given also in Croll's Climate and Time, Chapter XXIII.

54. Pratt. In the following notice of Pratt's investigations we shall 

refer to the fourth edition of The Figure of the Earth (published by 

McMillan & Co., London and New York, 1871). as it is the more recent 

and formal treatise. Although in many respects a valuable text book, 

it is marred by some serious errors, one of which we proceed to point out.

Pratt's investigation is based on the foil owing erroneous proposition 

(see page 2L2, Figure of the Earth), namely :

To prove that the effect of a mass at the earth's surface, -whether above or below, 

is to make the sea level rise at any place through a space, , wliere Fis the potential of the mass for a point on the disturbed sea level, which is iu the same vertical 

line with the place.

By a process of reasoning to which there appears to be no objection 

he arrives at this equation, namely,

V

r+coust= , (a)

9

in which r is the radius-vector of any point on the disturbed sea surface, V the potential at that point of the disturbing mass, and g is the 

well known velocity increment due to the earth's attraction. The radius 

of the undisturbed surface, supposed spherical, being denoted by a, 

Pratt says:

Let r a where F=0 or the horizontal attraction, of the mass first becomes appreciable.

He thus finds in the above equation

coust= a, 

and hence

. . . v rise m sea level=r tf:~_. (ft)

9

(1550
72 FOEM AND POSITION OF THE SEA LEVEL. [BULL. 48.

tf

But this reasoning is strangely faulty. V is not zero for r=a-, in 

fact it is zero only for points infinitely removed from the surface under 

consideration. He should have reasoned thus: Along the line of intersection of the disturbed and undisturbed surfaces r=a. Call the particular value of V along this line F0 . Then we have from (a)

const= a+ °, 

9

and hence

r-a= V~ Vo . (c) 

9

This agrees with our formula (6)', and the constant V0 is to be determined from the condition that the disturbed and undisturbed surfaces 

contain equal volumes. Formula (a) gives results which are too great 

by the constant amount , i.'e., this formula measures heights above

* »/

a spherical surface ° below the undisturbed surface (see section 5).

9

TO, it will be observed, is never of a lower order than V and can not 

therefore be neglected in comparison with V.

Although Pratt makes use of the correct principle for determining 

the constant F0 in his article 199, he ignores this principle altogether 

in his article 200, referred to above, and again in his article 213.

55. In his calculation Pratt assumes a sheet of ice 7,000 feet thick at 

the pole to extend over a whole hemisphere, decreasing in thickness, 

however, as it recedes from the pole in the ratio of the square of the 

cosine of the polar distance. He does not consider the effect of the 

rearranged water. His method of determining the potential V is not satisfactory. It consists (see articles 90, 91, and 92, Figure of the Earth) 

in a species of mechanical quadrature, by which he computes five special 

values of the attraction of a " hemispherico-spheroidal meniscus," whose 

thickness varies according to the law stated above. From these five 

values he derives by the method of indeterminate coefficients a general 

formula for the attraction of the meniscus, and from this formula by 

integration he gets a general formula for the potential of the mass. 

The order of approximation of these formulas is not shown and is not 

evident. From a test we shall apply it is inferred that the approximation is so rough as to render the formulas worthless.

We may readily derive the proper expression for the elevation of the 

sea under the conditions assumed by Pratt from our general equation 

(83). In this equation, if the thickness at the pole be denoted by h0 , we 

have

=/*0 cos2 /?, A=o, A=f-

(150)
vvoornvAUD.] INVESTIGATIONS OF PRATT. 73

Therefore (83) becomes

,/=

The first term in the second member of this equation is , and the

*/

second term is of equation (c) above. Now, for the ratio of the density

»/

of ice to the mean density of the earth P- Pratt uses - , and hence the

Pm b 

constant by which his results should be diminished, if they were what

they purport to be, is

56. We may test the correctness of Pratt's formula for computing 

by deriving the elevation of the disturbed surface at the center of his 

ice sheet. For this point we have by equation (26)

The integral in (d), therefore, is

and (d) becomes

«"=IF*«-:!5*"

This assigns the height of the disturbed above the undisturbed surface 

at the center of the ice mass. Now,

-^.=0.18856,

J.O

(137)
74 FORM AND POSITION OF THE SEA LEVEL. [BULL. 48.

but Pratt's formula (see page 237, Figure of the Earth) gives 0.1189, which 

is only about 63 per cent, of the true value.1 By a fortuitous compensation of errors, however, Pratt's formula gives a nearly correct result 

for the elevation of the sea at the center of the ice cap, the error in the 

potential for this point being about equal to. the constant omitted. His 

formula gives for this elevation 0.1189/t0= 0.1189x7000 feet =832 feet. 

'The correct value is (0.18856 1\)/?0 =737 feet, which is about 13 per 

cent, of itself smaller than Pratt's value.

57. Heath. The investigations of Heath may be found in numbers 

CCVIII, CCIX, and CCXEII of the Philosophical Magazine. In number COVIII he develops the theory of the effect on the sea level 

of an ice cap of uniform thickness and any angular radial extent, 

and applies this theory to a numerical example. In number CCIX he 

corrects a blunder by which he was led in his previous paper to the 

conclusion that the ice cap would produce a rise of sea level in both 

hemispheres. His last paper in number CCXIII is chiefly interesting 

as a review and criticism of the work of Croll, Archdeacon Pratt, and 

Sir William Thomson.

The method followed by Heath is that of Laplace's functions (spherical harmonics). He takes account of the rearranged water on the 

supposition that it covers the whole sphere and is free to permeate the 

ice mass. He considers masses of uniform thickness only, arguing, 

though not very cogently, that this was the probable form of the ice 

cap. His mathematical processes are correct in principle, but his formulas defining the position of any point of the disturbed surface are 

rather uninviting to the computer on account of the slow convergence 

of the series used. His series is the same as that in the second member of our equation (66), which we have separated into the definite integral and rapidly converging series of the third member of (66).

1 That the first term in the second member of (d) 1 is for the point in question is

easily proved directly. Thus, to terras of the order we ueglecfc, the potential of the 

mass is

/? . /to cos2 j8 . r0 d(3 

"

2r0 sin tL

'00 cos "2 cos2 fid/3

Dividing this by g=^rQpm -n we get

ZL=3/i SL *^lA=?/JLli0 if -£-=1 

g " °/Jm ' 15 15 ° On 6

(158)
WOODWARD.] INVESTIGATIONS OF HEATH. 75

58. In his numerical example Heath assumes an ice sheet of uniform" 

thickness, 10,000 feet, to extend 30° in every direction from the pole. 

For the ratio of the density of ice or sea water to the mean density of 

the earth he uses £. That is, in the notation of our equation (66) he 

has

Ji=10,000 feet,

/?=30°, 

£.=£»=* .

Pm Pm <>'

Heath gives the position of the disturbed surface at the two poles 

and at points 35° distant from either pole. He uses fifty terms of his 

series. We will verify one of his results, viz, that assigning the elevation of the disturbed surface at the center of the ice cap. With the 

above data formula (66) becomes

At the center of the ice cap

I=n sin L ,

/, (cos '«)=!. 

The numerical values of the first; ten terms of

4*4-1

are, in order,

+0:02500, +0.00058, 

+ .01203, - .00008, 

+ .00061, - .00039, 

+ .00358, - .00047, 

+ .00172, - .00041.

The sum of these is

+0.04817.

Therefore the above equation becomes

0+ Ju=959 feet +120 feet = 1,079 feet.

For the same result Mr. Heath gives 1,078 feet.

59. In his review Heath first devotes some space to a criticism of the 

views of Oroll and the work of Thomson. Croll had apparently held 

the notion that the change in sea level clue to an ice cap is essentially

(159)
76 * FOKM AND POSITION OF THE SEA LEVEL. [BULL. 48.

"equal to the change in position of the center of gravity of the earth 

and cap. This notion is indeed correct for the particular ideal case considered by Croll and . Thomson, but it is in general quite incorrect. 

Heath points out this fact and concludes that the special case discussed 

by Croll and Thomson affords us an inadequate conception of the actual 

problem.

With reference to the investigation of Archdeacon Pratt, which hi 

been published in number CCVIII of the magazine, Heath says :

I must confess, with some diffidence, that it appears to me radically erroneous.

He then proceeds to mention, correctly in the main, but with some 

reservation, the defects of Pratt's method. On one point, however, he 

expresses some doubt as to the adequacy of his own process. He is 

not sure that the order of approximation of his expression for the potential of the ice cap is sufficient, and thinks this may require closer 

investigation. We have cleared up this point in Article IV.

60. Thomson. The contribution of Sir William Thomson to the discussion of this subject is appended in the form of a note to a paper by 

Croll, On the Physical Cause of the Submergence and Emergence of 

the Land During the Glacial Epoch, published in number CCIX of the 

Philosophical Magazine. The note is brief, but it contains a clear statement of the essential analytical considerations required in the solution 

of the general problem, and of the decided simplification which results 

when the ice sheet has the special form assumed for the purpose of dis- « 

cussion by Croll, namely, that of a hemispherical meniscus, whose thickness (or density) varies everywhere as the sine of the latitude. In 

analogy with the views of Croll, Thomson devises the following ideal 

conditions, which are interesting as presenting the mechanical features 

of the problem in their simplest form. He says:

As an assumption leading to a simple calculation, let us suppose the solid earth to 

rise out of the water in a vasfc number of small flat-topped islands, each bounded by 

a perpendicular cliff, and let the proportion of the water area to the whole be equal 

in all parts. Let all of these islands in one hemisphere be covered with ice, of thickness according to the law assumed by Mr. Croll, that is, varying in simple proportion 

of the sine of the latitude. Let this ice be removed from the. first hemisphere and 

similarly distributed over the islands in the second.

Thomson gives no analysis, but continues:

By working out according to Mr. Croll's directions, it is easily found that the 

change in sea level which this will produce will consist in a sinking in the first hemisphere and rising in the second through heights varying according to the same law 

(that is, simple proportionality to sines of latitudes), and amounting at each pole to

where * denotes the thickness of the ice crust at the pole, i the ratio of the density 

of ice, and iv that of sea water to the earth's mean density, and GO the ratio of the 

area of ocean to the whole surface.

61. We may readily get Thomson's result from equation (66). Thus, 

the change in position of any point of the sea surface will be, if we
WOODWARD.] INVESTIGATIONS OF THOMSON. 77

represent the thickness of the ice at any point whose polar distance is 

ft by h0 cos /?,

But from equation (46)

1 V / / -- fff*f\d /n \ Ql Tl /I = / ^ I i>Uo fJ j oil I tJ *

and by the theory of Laplace's functions

f{ (cos j3) cos /? sin /3d/3=Q,

except when *=1. In this case, si nee/(cos /?)=cos /?, the last integral 

becomes

Therefore, observing that /f (cos o-)=cos a, a being the polar distance 

of any point of the sea surface, we get

' . 1 __ W 

Pm

Now, since the ratio of the area of the ocean to the whole surface of 

the earth is assumed in Thomson's problem to be ft?, we must replace p 

in (e) by (1 ca)p and pw by copw . Making these substitutions, and putting <*=0, the second member of (e) becomes

!-<»=-<

pm

This is Thomson's result.

Knowing the fact expressed by (e), namely, that the transfer of such 

a meniscus as we are consideringt from one hemisphere to the opposite 

one would change the sea level at any point by an amount proportional 

to the cosine of the polar distance of that point, we may get the result

(161)
78 FORM AND POSITION OF THE SEA LEVEL. [BULL. 48.

{/) ky equating the sum of the moments of the transferred ice and 

water to the moment of the whole earth. In other words, this particular case requires only the amount of shifting' of the earth's center of 

gravity. Calling this amount GI , and taking a plane perpendicular to 

the axis of the meniscus at the disturbed center of gravity of the earth 

as moment plane, the equation of moments is

/ 7T

(1 GO)P I 27ZT0 sin 6 , rQ d6 . Ii 0 cos 6 . r0 cos 6 +

c/O

C* 4

copw I 27zr0 sin 6 . rdti . GI cos 6 . r cos 6=^5 7fr^pm ff\. Jo &

This gives

l-GO&Z

Pm

which is the same as (/). 

In his numerical example Thomson takes

/to=6,000 feet,

2

_P l 

pm 6

These data give 379 feet as the change in sea level at the pole during 

the interval between the epochs of minimum and maximum glaciation. 

This is the greatest change in sea level that could occur under the assumed conditions.

62. It is to be observed that the numerical results of Pratt(corrected), 

Heath, and Thomson are not directly comparable with each other nor 

with the results we Have computed in Article XIV, since they are all 

based on different data. They represent effects due to causes of the 

same kind, but of widely differing magnitudes. No statement of these 

results would be intelligible without an accompanying statement of the 

data on which they rest. To show clearly how widely different writers 

differ in their data the latter have been collected in a tabular form below. It will be remembered that Pratt took no account of the effect of 

the rearranged water : l

1 One of the most important contributions to the discussion of the effects of continental glaciers on the sea level has appeared since the manuscript of this paper was 

placed in the printer's hands, viz: Die Geoiddeformationen der Eiszeit, von Erich 

von Drygalski, Dr. Phil. W. Porinetter, Berlin, 1887.

(162)
WOODWAHD.] EFFECT OF CONTINENTAL MASSES. 79

, Table showing data used by different authors in discussing the problem of glacial

gence. *

Author.

Pratt .................

Heath ...............

Ratio of 

density of 

ice to mean 

density of 

earth.

P_

i

i

i

Ratio of 

density of 

sea water to 

mean density 

of earth.

Pm

i

2

2

Angular 

radius of 

ice mass.

A,

0

90

30

. 90

no

Thickness of ice at 

angular distance /3 from 

its axis.

Feet. 

7,000 cos2 /3

10, 000 uniform.

1 Bin»lflJ

XVI. VARIATIONS IN SEA LEVEL ATTRIBUTABLE TO CONTINENTAL

MASSES.

63. To illustrate the use of the theory developed in determining the 

disturbance of the sea level attributable to a continental mass we shall 

consider, in addition to some observations on the general features of 

the subject, the special case presented by the largest of the continents, 

namely, Europe and Asia.

In contemplating this problem it is important to distinguish two extreme hypotheses relative to the nature of the earth's crust. On the 

one hand, we may suppose that a continent is simply a superficial aggregation of matter, which, if removed, would leave a sensibly centrobaric spheroid j in other words, the presence of a continent does not 

imply that beneath it the earth's crust is any less dense than beneath 

the ocean. On the other hand, we may suppose that the several radial 

element prisms of the earth's crust are in a state approximating to hydrostatic equilibrium, and hence, that the mere existence of a continent 

implies a defect of density in the strata beneath it. According as we 

proceed from the one hypothesis or the other we shall arrive at widely 

differing results, which may be regarded, however, as the limits between 

which the facts lie.

64. Assuming, in accordance with the first hypothesis, that the forma, 

tion of the continent in question involved a transfer of the earth's 

center of gravity towards the center of the continent, it will be of interest to compute the position of the disturbed sea surface with respect 

to spherical surfaces of the same radius and concentric about the original and disturbed centers of gravity, respectively. For this purpose 

we may use formulas (64) and (79). These are based on the assumption 

that the attracting mass is of uniform thickness. A more reasonable 

assumption is, perhaps, that the continents slope up rather rapidly, but 

not abruptly, from the sea shore, attaining their average height at no

(163)
80 FORM AND POSITION OF THE SEA. LEVEL. [BULL.48.

great distance inland; and more accurate knowledge than, we now 

possess might render it desirable to use some form of the more general 

equation (83), which takes account of variations in thickness of the 

attracting mass. As we can not expect, however, at present, to represent the actual shape of a continent very closely, and as we shall not 

attempt to estimate the attraction of the rearranged free water on itself, 

it will be best to confine attention to masses of uniform thickness, which 

give effects greater in general than the probable actual effects.

65. For the angular radial extent and the relation of thickness and 

densities for the continent of Europe and Asia we take the following 

data substantially as they are given by Helmert in his Geodasie1 , Part 

II, pp. 313, 314, viz:

/?=arc of 38o, 

3/*p/pm =4,000 meters 

=13,124 feet.

The last expression is arrived at by taking for the average depth of 

the sea 3,438 meters, and for the average elevation of the continent 440 

meters, the density of the continental mass being assumed to be half 

the earth's mean density, or 2.8, and that of sea water 1. Thus we 

have a mass 3,438 meters thick, whose effective density is 2.8 1=1.8, 

and an additional mass 440 meters thick of density 2.8. These two are 

equivalent, so far as their potential to terms of the order we neglect is 

concerned, to a single mass of density 1.8 (or £pm) and 4,120 meters 

thickness. Hence, in round numbers, the relation above.

The value for the angular radius /3 of the continent is equivalent to 

about 2,600 miles, measured along the surface of the earth. It is the 

same radius assumed in Article XIV for the ice mass. This value is 

also very nearly that angular extent which a continent of uniform 

thickness must have to produce the maximum upheaval of water 

along its border (see section 23).

The position of the disturbed surface relative to the two spherical 

surfaces concentric with the original and disturbed centers of gravity, 

respectively, will be given at intervals of 10° from the center of 

the continent to the point opposite, or 180° from that center. The

requisite values of I /n sin '-~ for this purpose have been computed

2i (

from formulas (70) and (71), using the logarithms of git g^ etc., fcj, fe2 

etc., given on page 69. To determine the slope of the disturbed surface 

with respect to the spherical surfaces of reference (or the deflections of 

the plumb line) at intervals of 10° from the center of the continent, 

use has been made of equations (73) and (80), the differential coefficients 

dl /da being computed from (74) and (75). Omitting the details of 

the computation, the nature of which is readily apparent from the

1 For lull title see page 86.

(164)
WOODWARD.] , EFFECT OF CONTINENTAL MASSES. 81

equations referred to, the results are embodied in the following table. 

The first column of the table gives angular distances increasing by increments of 10° along a great circle from the center of the continent. 

The second and third columns give for each angular distance the elevation 

or depression of the disturbed surface relatively to the spherical surfaces concentric with the undisturbed and disturbed centers of gravity 

of the earth, respectively, elevations being indicated by the plus sigu 

and depressions by the minus sign. The fourth and fifth columns give 

for each angular distance the deflections of the plumb line or inclinations of the disturbed surface to the spherical surfaces of reference. 

The signs of these deflections are minus or plus according as the anglebetween the plumb line or normal at any point of the disturbed surface 

and the axis of the continent is greater or less than the angle between 

the radius vector of the same point and the axis of the continent (see 

Fig. 7).

Table showing the superior limiting effects in disturbing the sea level attributable to thecontinent of Europe and Asia.

Angular 

distance from 

center of 

continent.

0

00

30

20

30

, 40

50

60

70

80

90

100

110

]20

130

140

150

160

170

180

Elevation or depression of disturbed surface with respect to 

Spherical surface concentric 

with undisturbed center' 

of gravity 

of earth.

Feet.

+2, 881

+2, 812

+2, 588

+2, 148

-1-1, 099

+ 444

+ 111

- 104

 '255

- 365

- 448

- 512

 561

 600 '

 629

- 650

- 665

- 673

 676

Spherical surface concentric 

with disturbed center 

of gravity 

of earth.

Feet.

+1,637

+1, 587

+1,419

+ 1,072

+ 14.6

 356

 511

 529

- 471

- 365

- 232

- 87

+ 61

+ 200

+ 324

+ 426

+ 504

+ 552

+ 568

Inclination of disturbed surface with reference to 

Spherical surface concentric 

with undisturbed center 

of gravity 

of earth.

//

 0.0

-7.7

-17.7

 33.9

 61.3

-24.2

" -14.7

-10.0

 7.2

 5.4

- 4.1

 3.2

 2.4

 1.9

 1.4

- 1.0

 0.7

- 0.3

- 0.0

Spherical surface concentric 

with disturbed center 

of gravity 

of earth.

//

- 0.0

- 5.6

 13. 5

 27.7

-43.5

-14.8

- 4.1

+ 1.6

+ 4.9

+ 6.9

+ 8.0

+ 8.4

+ 8.2

+ 7.5

+ 6.5

+ 5.1

+ 3.5

+ 1.8

+ 0.0

Bull. 48 6

(165)
82 FORM AND POSITION OF THE SEA LEVEL. [BULL.48.

66. The relations of the disturbed surface to tbe spherical surfaces 

of reference are shown in Fig. 7. The full-line circles in. this diagram 

represent great circles of the spheres of reference on a scale of 2T'o<roFoo"o 

The position of the continent is indicated by the arc AB of the circle, 

whose center C is the undisturbed center of gravity of the earth. The 

disturbed center of gravity, or the center of the other circle, is at C'. 

The position of the disturbed surface is indicated 'by the dotted line. 

The radial distances of this line from the circles of reference and the 

distance between the centers G and C' are exaggerated 2,500 times. 

QPR represents a normal to the disturbed surface at P.

67. For the elevation of the disturbed surface at the immediate border 

of the continent we find from the second of equations (G7) and from 

(79) 1,380 feet or 400 feet, according as the sphere of reference is concentric with the undisturbed or with the disturbed center of gravity of 

the earth.

68. It should be remarked in this connection that the results given 

in the preceding table, and, indeed, all those in this section pertaining 

to the continent of Europe and Asia, are such as would exist under the 

sissumed x conditions if there was no other continent. The complete 

problem requires the determination of the resultant action of all the 

continents at any point of the sea surface. For a method of determining this resultant, when the components due to the several continents 

are known, the reader may be referred to Helmert's Geodasie.
WOODWARD.} EFFECT" OF CONTINENTAL MASSES. 83

69. To determine the deflection of the plumb line at the level of the 

sea along the border of the continent, we observe that according to 

our assumed data the portion of the continental mass lying below the 

sea level is about 11,000 feet thick and has an effective density of 1.8, 

while the portion above the sea level is about 1,400 feet thick and has 

an effective density of 2.8. We may compute the deflection of the 

plumb line due to each of these portions by means of equation (77) and 

add the results to get the total deflection. For the lower mass we have 

for use in (77)

h= 11,000 feet, p=18, 0=11,000 feet; 

and for the upper mass

/z=l,400feet, p=2.S, - % 0=0 feet,

Hence the deflections due to the upper and lower masses are 138" and 

34", respectively, and their sum is 112". This deflection is relative to 

a radius drawn to the undisturbed center of gravity of the earth. 

The deflection relative to a radius drawn to the disturbed center of 

gravity is by equation (80) 8" less, or 164".

70. Under the conditions of the second hypothesis, which supposes 

the several radial element prisms of the earth's crust in a state bordering on hydrostatic equilibrium, it is evident that the disturbances of 

the sea level attributable to a continental mass must be of a low order. 

They must, in fact, be confined to terms of no higher order than those 

which have been neglected in our equations defining the position of the 

disturbed sea surface. The precise evaluation of these terms would offer 

difficulties practically insuperable in all cases, except those which presentN 

the simplest arrangement of densities in the element prisms of the earth's 

crust. We may form a sufficiently definite idea of their smalluess, 

however, by considering the ideal question of the effect on the sea level 

of the radial transfer of a stratum of the earth's crust from some position below to some position above the sea surface. 

 Let the stratum considered.be a portion of a spherical shell; let its 

border be circular and of angular radius /?. The effect on the sea level 

will obviously be greatest at the axis of the stratum. Hence, we only 

need to derive the change in potential at the sea level and at this axis 

due to the transferred stratum. Suppose the stratum raised to a height 

lii above sea level. Let its uniform thickness in this position be ti and 

its uniform density pt . Then, r being the radius vector of any element 

of the mass and r' the radius vector of the sea surface, the potential 

of the stratum at its axis and at sea level will be

/

2n-'cos0

r'+hi+ti s# . 

rhlr I sm 

hl Jo vV+r'2 2

Sir'+tii + t, ______________ '

2pi?f I v/r2 -+r/2 -rr'cos//-(r-r/ ) |~.

t/r'-fAi -" T
84 FORM AND POSITION OF THE SEA LEVEL. [BULL. 48.

This to terms of the first order inclusive is

If we suppose the stratum in its original position is below the sea 

surface a distance 7^2 , and that its uniform thickness and density in this 

position are t2 and p2 , respectively, the resulting potential at. the same 

point of the sea surface is

(98)

Since piti=p2 ^ i- e., since the mass of the stratum in the two positions is the same, though its thickness and density may vary, the difference of (97) and (98) is

4r'sm£

^j

This expression shows that a stratnni of the earth's crust might be 

transferred through a considerable distance radially without materially 

affecting the sea level.

Thus, for example, suppose

^=7*2=10,000 feet, " £,=#2 =5,000 feet.

Then the fraction in the parenthesis of (99) becomes yio- Iu other 

words, the potential at the sea surface under the center of the stratum 

would be only ^ as great as that due to an uncompensated external 

stratum or internal vacuity of equal effective mass. 

ID the case of the continent of Europe and Asia we have (section 69)

7tj=0, fl=l,400 feet, p1= 2.8. 

These in (97) give

sin £ 1400x2.8. -, 20000 COOOOsiu Ai

Likewise for use in (98) we have

7t2 =0, #2=11,000 feet, p2 =l.S. 

These give

4r'Train £ 11000x1.8/1 2 \ 2o50 7640 sin g-2i

(168)
WOODWARD.] LIST OF AUTHORS. 85

The sum of (a) and (6) reduced to unit density is

8in>i23720f I- * -

3000 9000sill £

2i

L\ 

sill |^

This is the potential to terms of the first order, inclusive of the whole 

 continent at its center and at sea level, on the supposition that there 

is no defect in density of the underlying strata. But if we suppose a 

defect in density of 0.237 uniformly distributed through a depth of 

100.000 feet below the sea level, we must diminish the result (c) by

4r '7r^23720/l-JL_. . __. _

2 V 840 sin I

The remainder is

±r'7t i 23720

or .about 3-^ of the potential due to the uucompensated mass. Under 

these conditions of compensation, therefore, the elevation of the sea at 

the center of the continent of Europe and Asia would be only about 10 

feet, whereas the first hypothesis would require an elevation of about 

2,900 feet (see section 65.)

71. Notwithstanding the feeble effect a continental mass, whose radial

 elements are in a state approximating to hydrostatic equilibrium, would 

have in elevating or depressing the sea surface, it is conceivable that a

 considerable deflectiou of the plumb line might be produced by such a 

mass along its border. If, for example, we suppose the visible mass of 

the continent of Europe and Asia to be compensated by a defect in 

density of 0.18 uniformly distributed through a depth of 135,000 feet 

below sea level, the deflection of the plumb line along the border of this 

continent would be diminished by 117", leaving still a deflection (section 

69) of 172" 117// =55// relative to the undisturbed sphere of reference. 

This deflection, 55", it should be observed, is the maximum possible 

value under the assumed conditions. If the degree of compensation assumed actually exists, it is probable that the real maximum deflection 

of the plumb line is much less than 55", since our calculation premises 

a vertical coast wall for the continent, whereas it presents for the most 

part only a moderately steep slope along the sea shore.

XVII. LIST OF AUTHORS CONSULTED.

72. The authors whose works have been specially consulted in the 

preparation of the preceding pages, and the full titles-of their works, 

dates of publication, etc., are named below :

Bruns, Dr. Heiiiricb. Die Figur der Erde. Publication des konigl. preussiscben 

geodiitiscben Institutes. Berlin, 1878.

(169)
86 FORM AND POSITION OF THE SEA LEVEL.

Clarke, Col. A. Rf Geodesy. Oxford, 1880.

Fischer, Dr. Phiiipp. Uutersuchuugeniiberdie Gestaltder Erde. Darmstadt, 1S68.

Helmert, Dr. F. R. Die mathematischen und physikalischen Theorieen der hohereo 

Geodiisie. II Teil. Leipzig, 1884.

Laplace. Me"canique Celeste. Tome 3. Paris, 1878.

Pratt, John H. A Treatise on Attractions, Laplace's Functions, and the Figure of 

the Earth. Fourth edition. London and New York, 1871.

Stokes, G. G. Mathematical and Physical Papers, Vol. II, especially the paper On 

the Variation of Gravity at the Surface of the Earth. Camhridge University Press,. 

1883.

Thomson and Tait. Treatise on Natural Philosophy. Vol. I, Part II. Camhridge, 

1883.
INDEX.

Page.

Agassiz, slopes near Lake........................... G7

Annals of Mathematics, mathematical features published in................................... 13

Application of theory, to relative position

of level surfaces in a lake basin......... 58.

to effect of continental glaciers.............. CO

to effect of continental masses ............. 79

Attraction,

of disturbing mass ............................... 19

of rearranged free water....................... 35

of mass in a lake basin.......................... 58

of continental glaciers........................... GO

of continents.......................................... 79

Authors consulted, list of.......................... 85

Bonneville, level surfaces in Lake........... 17

Center of gravity of earth and disturbing

mass.................................................. 51

Center of surface of reference .................... 19,51

Chamberlin, Prof. T. C.,

request for mathematical treatment of 

glacial problems................................ 13

memoir on Driftless Area cited............ 17

Clarke, A. R.,spheroid of........................... 19

Constants,

of Clarke's spheroid...... ...................... 19

of sphere of reference..................... ..... 19

Foand U$, evaluation of....................... 37

Croll, James,

on alternation of glaciation.................. 60

on glacial submergence ....................... 71

views of, on effect of iee_mass............... 75

Data for computation of form and position

of level surfaces in lake basin........... 59

for effect of continental glaciers........... 61,79

for effect of continent Europe and Asia 80

Deflection of plumb-line, formulas for........ 47

at border of lake basin.......................... 60

at border of ice mass............................. 68

due to alternation of glaciation ........... 70

due to continent of Europe and Asia... 83,85

Deformation of sea-level or geoid (see disturbed surface), von Drygalski on..... 78

Density, effective in lake basin.................. 58

of earth's crust...................................... 58,83

of earth, mean....................................... 59,6!

of ice...................................................... 61

effective in case of continent Europe 

and Asia.......................................... .. 80

Dimensions of earth's spheroid.................. 19

of assumed ice masses.......................61 et seq.

of continental mass of Europe and Asia.. 80

Disturbed center of gravity of earth........... 51

> Page. 

Disturbed surface, defined.......................... 19

equations of..................... ....... ............ 40

slope of....................".... ........."................ 47

Driftless area, Chamberlin on the.............. 13,17

Drygalski, Erich von, memoir on Ceoiddeformationen der Eiszeit cited......... 73

Effect of re-arranged free water.................. 35

of removal of water in lake basin......... 53

of continental glaciers... ........ .............. GO

of continents on sea-level..................... 79

Elevation of disturbed surface.................20,41,42

Ellipsoid of reference, Clarke's elements of.. 19 

Equationsof disturbed surface .................. 40

for disturbing mass of uniform thickness........................'............................. 40

for disturbing mass of variable thickness..................................................... 53

Equipotential surfaces, defined................... 12

in lake basin ..................... ................... 53

Europe and Asia, continent of.................... 79-85

Evaluation of potential of disturbing

mass................................................... 21

of constants F0 and Uo.. ....... ................. 37

of definite integrals Jj, I'i...... ................. 43

Fischer, Philipp, treatise of........................ 86

Form, general of sea-level.......................... 15

of disturbed surface .............................. 41

of level surfaces in lake basin............... 53,59

of sea-level as disturbed by assumed 

continental glaciers........................... 66,70

of sea-level as disturbed by a. continental mass ....................................... 81,82

Free water, attraction of............................. 3~>

Geodesy and geology as related to form

and position of sea-level..................... 15

Geoid, position of, relative to spheroid...... 15

deformation of, von Drygalski on........ 78

Gilbert, G.K., request of, for investigation.. 13 

problem proposed by ........................... 17

on slopes of lake beaches in New York.. 67 

Glaciers, continental, influence on sealevel................................................... 60

assumed shape of................................. 61

slope near border of.............................. 62

assumed volume of .............................. 64

assumed mass of.................. ................ 65

deflection o/ plumb-line on border of.... 68

Glacial submergence

discussion of,by Croll........................... 71

discussion of, by Pratt....v..................... 71

discussion of, by Heath.......................... 74

discussion of,-by Thomson.................... 76

87

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88 INDEX.

Page. 

Glacial submergence Continued.

discussion of,by von Drygalski........... 78

table pf data used by different authors 

in discussion of................................... 79

Graphical illustration,

of level surfaces in lake basin............... 59

of shapes of assumed ice masses........... 03

of disturbed surface due to alternation 

ofglaciation ...................................... 70

of disturbed surface due to continental 

masses............................................... 82

Greenland, thickness of glaciers in............ 03

Heath, D. D., on glaci al submergence........ 74

criticism of Croll and Pratt.................. 75

Heine, E., treatise cited.............................. 33

Helmert, F. R., treatise cited..................50,80,82

Hypothesis relative to form of continental

glaciers............................................... 01

relative to nature of earth's crust......... 79

Ice sheets, assumed slopes of surface and

thickness of................................ ...... 01

assumed volume of............................... 04

Illustrations, graphical, of level surfaces in 

. lake basin.......................................... 59

of shapes of assumed ice masses .......... 03

of disturbed sea surface-due to continental glaciers.............."...................... 70

of disturbed sea surface due to continents................................... ............... 82

Key to symbols used in this paper... ........ 9

Lake Bonneville, beaches of......................." 17

Lake Agassiz, beaches of........................... 07

Lake basin, position of level surfaces in... 10.58 

Lake Ontario, beach slopes near............... 07

Laplace's functions.......... .......... ............... 30

Laplace, demonstration cited..................... 30

Level surfaces, defined................................ 18

in lake basin......................................... 58

slope of, in lake basin .......................... CO

List of authors consulted ........................... 85

Mass, of earth.......... ........ .......................... 19

disturbing, denned................................. 20

in assumed ice-caps............................... 61

Mathematical statement of problem of this

paper.................................................. 18

Mathematical symbols used in this paper,

key to............. ....^............................... 9

McGee, W J, on maximum synchronous

glaciation............................................ 03

Page.

Nordenskjokl on slope at border of Greenland ice-fields..... .............................. 63

Ontario, slopes of beaches near Lake......... 67

Plumb-line deflections, formulas for.......47,51,52

at border of assumed lake basin ........... 60

at border of assumed ice mass............... 08

Potential of disturbing mass....................... 19

expressions for...................................24,33,35

special values of................................... 25

- degree of approximation of.................. 20

Position, relative, of disturbed surface........ 19,41

of level surfaces in lake basin............... 58

Pratt, Archdeacon J. H., on glacial submergence.........................................71-74,79

Price, Bartholomew, treatise of, cited ......... . 27

Slope of disturbed surface, formulas for...47,51,52 

Slope of upper surface of assumed ice

masses............... ................................ 01,03

Slope of glacial lake beaches.................... .. 67

Sphere of reference..................................... 19

Spheroid of reference ......;.......................... 15

Clarke's elements of.............................. 19

Table of values showing

relative position of level surfaces in 

lake basin .......................................... 59

slopes of assumed ice masses ............... 02

thickness of spherical shells of equal 

volume with assumed ice masses and 

equivalentloweringof sea-level......... 04

position of disturbed surface in case of 

assumed ice masses ........................... 06

minimum thicknesses of ice masses producing steep border slopes of disturbed surface.....;.... ......................... 08

disturbance of sea-level due to assumed 

ice-cap................................................ 70

superior limiting effects, on sea-level attributable to continent of Europe 

and Asia................................. ........... 81

Thomson, Sir W., on glacial submergence ............... ...............................71,76,79

Undisturbed surface defined ....................... 19

Upharn, Professor Warren, on beaches and

terraces of Lake Agassiz.................... 67

Variations in sea-level attributable to continental glaciers................................. 60

attributable to alternation ofglaciation.. 69 

attributable to continental masses........ 79

Works referred to ........................ ........ ..... 86

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