Lancaster daily intelligencer. [volume] (Lancaster, Pa.) 1864-1928, 1886-11-13

by looking At this figure 1t becomes a,
once apparent that some changes are neces,
sary TO make il suit the Quantities used by
Newton. Newtnn's J. ~ evidently taken n
radius OF his circle. Therefore the value OF
to radius K OF the functions must he taken
out ana |. the radius Of the circle introduced.
To radius unity the formula becomes
81 since yer sins was taken to radius 4.
Since Newton + is TAX Ell to radius l. the -
If h
X ~
The general value of the force -
Sm.,n',. from Newton's demonstration.
Now suppose L the place OF tho body to he at
point where the chord passes through both
and o which must happen twice for every
revolution OF L. Then II is evident that for
one instant PN-NA or the chord equals the
diameter. Then for that point VA may
cancel two factors of PN . and leave PY.
This leaves Newton's F-,''--,,.. But PV 60
81, 1t being the diameter OF the circle for the
moment and SP-r. Therefore F." pre-.
the work.
The meaning OF all This is that there is
really no difference between the results for
the cases In which both apply. The result
after the pattern OF the new theorem consist
ently carried out Is only "A special value at A
and /. OF the figure belonging to IL and
agrees exactly with Newton's result for the
same points. The result Of the new thsOrem
Il not general, because it does not treat the
general case of taking the radius vectOr In
~ hFa hn
demonstration Is somewhat deceptive In this
respect While the radius vector NAV TO
taken in any direction Tor the sake OF deriving
the equation, In its limiting position 1t must
always come back TO A and thus the force ~
computed for A and that only. The general
case is represented by Newton In his figure
when SP is the veCtOr and II there was any
occasion to use an elementary triangle its
limiting position ought TO be taken at L, any
point not simply at the ends OF the diameter
through . and y. lt Is evident that the
yer SIN. can only represent the centri petal
force at A the end Of the diameter through b
and OF Zig. belonging to tho new theOrem
or q and ) OF Newton's figure. Toshow that
in general the Z YER sin, cannot represent the
centripetal force We need only consider tho
direction OF SP as given in Newton's figure.
The centripetal force acts from Z to s while
the YER sin. would extend along line from L
To q. Therefore the versn. here does not
represent the centripetsl force.
The difficulty about this whole matter lies
In The fact that the author errs in supposing
that he has the general case when he has onl v
particular case, or a solution for a point at
the ends OF the diameter through the centre
OF the circle and the centre OF attraction. II
VIII not do TO conclude a general law from n
special case since, here, as clearly shown
above, sometimes factors may disappear or
rather will fail to be taken into account and
no true law can he the result.
Thus we have interpreted the new tbeorem
with somewhat changed result to its bes,
advantage and have reconCiled the result will,
Newton's result for the corresponding case.
Ihe only conclusion that can be reached Is
,hat US no theorem at all; but simply prob
not matter what other theorems NAV follow
| 1t cannot change the particular case into I
| general onei Tor as soon IN the point p leaves
| the diameter through and s the form o.
demonstration OF the new theorem cannot bt
| used.
| SCHOLL OBSERVATORY,