1’:(.f !

CIC-14 REPORT COLLEC~ON

FMPRODUCTION

COPY

LAMs-2421

LOS ALAMOS SCIENTIFIC LABORATORY

OF THE UNIVERSITY OF CALIFORNIAo LOS ALAMOS NEWMEXICO

I

THE PENETRATION OF RADIATION

WITH CONSTANT DRIVING TEMPERATURE

—

..

—.- i:’.

.... ..... .

,..

.. . .,.,+., <,”:-

.,

.,

.;-

.-, . .-,.

.,- 7.* <. . .

.,

..

... ,.

ri+o... ..ti.+.., . .-.>w–.”.2 ,8 :,.... ‘. ..] : L-.

., h...--,.,...,!’ . .. . . ..... ..2 !-. I

F’. ..-

,.

.. ....

..

-,.,.. .’.

,,,; ,. -

. .

... ,’

— -.1.

.

-. .

-..

,.. :’

..-.L-= : --- -

.. ---- . -,, ,.+ ,’. I

}, * ,.;

,“! ..: t -: #- ~;< $.:,.

.“ I s% ~,’..:,—.- .– .. .. ... . .,, . . ... -.’ .. ..,,,...,,, +,, -. . . ... ....’ +- k.. -1

. . .. . .. .. .. . .. .. . . . . .. .... ... .-, . ..., ,,, ,. .

CA. ----- ,. k

..,.-..

. . . . . ..

. ..-. -.$ . . . . . ---

--lr’-w .“”.. - ‘:””’ . ““‘“. ---

. ...,

----- :.. .

.

.. . . .:. .

. . .. .

1

.... ~,.y. :.

1. ‘.

----- . I
‘t , r.

LEGAL NOTICE

This report was prepared as an account of Government sponsored work. Neither the United States, nor the

Commission, nor any person acting on behalf of the Commission:

A. Makes any warranty or representation, expressed

or implied, with respect to the accuracy, completeness, or

usefulness of the information contained in this report, or

that the use of any information, apparatus, method, or process disclosed in this report may not infringe privately

owned rights; or

B. Assumes any liabilities with respect to the use

of, or for damages resulting from the use of any information, apparatus, method, or process disclosed in this report.

As used in the above, “person acting on behalf of the

Commission” includes any employee or contractor of the

Commission, or employee of such contractor, to the extent

that such employee or contractor of the Commission, or

employee of such contractor prepares, disseminates, or

provides access to, any information pursuant to his employment or contract with the Commission, or his employment with such contractor.

Printed in USA. Price $1.00. Available from the

Office of Technical Services

U. S. Department of Commerce

Washington 25, D. C.
1

-—.

I

.-

LAMS-2421

PHYSICS AND MATHEMATICS

(TID-4500,15thEd.)

LOS ALAMOS SCIENTIFIC LABORA~RY

OF THE UNIVERSITYOF CALIFORNIA LOS ALAMOS NEW MEXICO

REPORT WRITTEN May 1960

REPORT DISTRIBUTED JUIY 29, 1960

THE PENETRATION OF RADIATION

WITH CONSTANT DRIVING TEMPERATURE

Work Done By:

Albert G. Petschek

Ralph E. Williamson

John K. Wooten, Jr.

Report Written BY

AlbertG. Petschek

RalphE. Williamson

ContractW-7405-ENG. 36 withtheU. S.AtomicEnergyCommission

All LAMS reports are informal documents, UEually prepared for a epecial purpoee and prfmarlly prepared for uee witbln the Laboratory rather than for

geneml dfatrIbution. This report bas not been edited, reviewed, or verified

for accuracy. All LAMS reporte express the views of the authore aE of tbe

time they were written ad do net necessarily refleet the opixdom of the Loe

Alamoe SoientLfio Laboratory or tbe ffnal opinion of the authore on the subject.

-1-

.-

ABSTRACT

Exact and approximate solutions for the penetration of radiation

with constant driving temperature are presented.

-3-

1. Scope. We investigate the diffusion equation for radiation under

the following assumptions:

(a) no energy in the radiation field and Cv . constant

(b) constant density

(c) an opacity K = ICopaT-n

(d) constant driving temperature

(e) plane geometry.

Many of these assumptions can be generalized or mitigated with some additional complication in the arguments. In only one place is there any

real difficulty in generalizing (a) and (c), and this is in the homology

transformation (see Section >). It is possible to make (b) more reslistic in some cases.

*

And, as Marshak pointed out a number of years ago,

(d) cknbe generalized to cases where T atp orT aept. However, we

shsll restrict ourselves for the moment to discussion of the tiffusion

equation with limitations (a)-(e). We have a two-fold purpose in mind:

to provide exact solutions for cases of different n and to elaborate a

consistent set of analytical approximations to the solution of the diffusion equation. In our experience, assumptions (a)-(e) are remarkably

*R. E. Marshak, Los Alamos Scientific Laboratory Report LA-230, February

194.5.

-5-
good for many

can be forced

curacy of the

classes of problems, and in any case, most radiation codes

to obey these assumptions so that questions as to the accede may be settled. With a view to satisfying the two

puwoses mentioned, we shall: first, make use of the well-known similarity transformation to reduce the partial differential equation to an

ordinary differential equation; second, tabulate exact solutions; third,

systematize the hierarchy of “constant flux” approximations; and finally,

tiscuss the accuracy of the first few of these.

We do

of what we

to make it

not claim originality for this work. On the contrary, most

are setting forth

readily available

has been done many times; our aim is simply

once and for all.

2. The similarity transfomnation. In general, the radiation diffusion

equation is

where E is the internal energy

note, moreover, that the flux,

?=-ac&T4

T ICP

(1)

per unit volume of the medium. We may

+

F, is givenby

(2)

and tiat the total internal energy of the wave is

‘tot = J

Edv,

-6-

(3)
where the range of integration is over the whole volume of tie radiation

wave.

Under our assumptions, the diffusion equation becomes

where x is a geometrical variable. Thus

where

tion

K

ac 1 =—0

3 2ia” E+-c”

‘oc#

The expressions for

wave (expressed per

aT4+n

F= -@cv “~

(4)

(5)

(6)

the flux and for the total energy in the radiaunit normsl area) become

(7)

and

x

J

o

‘tot = Pcv o

T@ (8)

where x . 0 is the point at which the constant driving temperature is

applied, and x = X. is the head of the wave. It may be remarked in

passing that the often-used quantity P} the radiation potential,is

-7-
defined for a geometrical (as opposed to a mass) coordinate as

P= c@&+4 .

Now we define the similarity variable ~ for t >0 as

It is easy to verify that

(9)

(lo)

(11)

At the ssme time, we find it convenient to express the tqperature as a

dimensionless variable, T:

T(x,t)

r=

To $

(12)

where To is the constant driving temperature. If we make use of Ms. (11)

and (12) in Eq. (5), we obtain

(13)

-0-
where we have set

m= n+h

1

In these new variables, Eqs. (7) ad (8) for the flux and the total

energy in the wave become

and

(14)

(15)

(16)

with

where go is that vslue of ~ which is obtained by substituting X. and t

in the definition (10) for ~. The condition for energy conservation,

namely,that

is easily seen to be equivalent to

-9-
which can be obtainc

g=o

‘

d by differentiating Eq. (16) with

and comparing the result with Eq. (15).

3. Boundary conditions. It seems physically obvious,

(18)

respect to time

it is borne out

by experience, and it has been shown by Marsh~ that (excePt for the

*

pathological -- because linear -- case, m = 1) the boundary con~tions

which insure that a solution

(a)~=latg=O

(b) at some point go,

The first

contition

flux = 0,

follows fmm the

which determines

simultaneously.

T

of Eq. (13) be

definition Of T,

the front of the

a true radiation wave are

(l$la)

(l$?b)

Eq. (12), the second is the

WV% where T(go) = O ~d the

“The case m = 1 is the well-known heat conduction equation. Here the

boundary condition (lgb) is inapplicable and is replaced by one reqtiring that T +0 as ~ +00. The relevant solution is

‘1-=1 -@@(5/w),

where

J

x

o(x) = ~

e-t*dt

Fo”

The energy in the wave is still finite, and in the dimensionless variables is 2/fi.

-1o-
k. Formulation of the equation for numericsl integration. Near ~ = O,

Eq. (13) looks like (d2Tm/d~2) = O and, therefore, Tm is linear in ~.

Since T is finite, all powers of T are well-behaved near the origin.

Near ~ = ljo,Eq. (13) is approximately

(20)

and its asymptotic behavior may be found from the indicial equation.

Let

‘= [%- r’p+● 00●

Then Eq. (20) gives, to lowest order,

-~ :A1/p

Op (50- J1/P)-l =:(:- l)Am/p(~

o-

&/P)-2a

From this it follows that

P =m- 10

This satisfies (d’rm/d~) = O, which requires only p <m.

Since T is now zero, powers of T greater thau the pth snd the ~

axis osculate more and more closely, while powers smaller than the pth

have a singular first derivative. A numericsl integration will therefore proceed most conveniently after a change of variables. We put

-11-
m-1

P=T s (21)

Using this in Eq. (13), we obtain the following form, which we use for

the numerical integrations:

5. Homology trsnsfoxmation.*

1

(22)

The search for go will be facilitated

observing that Eq. (22) is invariant to the following transformation.

Let us set

in Eq. (22) and find the conditions

same form ti X,@ as it was in ~,qo

And SO

by

(23)

on a and @ which render it of tie

Now

*The arguments of this section were called to our attention by Peter

La% .

-12-
Then if

a2 = 8)

Eq. (24) becomes

[

~lt= Q’ ~+ m

- iim -0’ 1,

nl-

(24)

(25)

(26)

which is i.denticsl in form to Eq. (22).

~is enables us to meet condition (lga) given an arbitrary solution.

For;suppose that we have one integration which progresses backward from

$1) (1) (2) and crosses the q-axis at PO , what should we use for go to

alive at q(’) =.?

o

-13-
In Eqs. (23), identify @,x with the first trisl values of q and ~

and the left-hand sides with the second:

J2)

o

= ~ E:)

J2)

o

=139y.

The second of these gives

$=#T”

o

But from Eq. (25),

and

$2) =

o

1

r

----@

o

p.

o

(27)

6. Numerical.solutions. Equation (22) has been integratedby a fourth

order Runge-Kutta method for

~= 0(0.5)5(1)10

In= 4(0.5)9(1)14.

-14
For each m, two integrations were performed, the first from ~. = 1 backward to ~ = O. After the first integration, the homology transfomnation

(27) was used to find that go which would satisfy the boundary condition

on (p at the origin.

The step interval for the integration was taken to

except for the first step,which was -A~ = go mod( .05).

A~ = -0.1 andA~ = -.01 showed no

icant figures.

Initial values for q’ and q“

numerical differences

be A~ = -0.05

Integrations at

to six signif-

(this because of its indeterminacy at

E = ~o) are needed to s-tart the integration. These are obtained from

the Taylor series near ~ = go:

~=@(Eo - E)+* @’(Eo- 5)2+

Inserting this into Eq. (22), one obtains

.*.*

therefore,

@’= - =&

-l’j-
In Table I and the figures will

grations.

7. The constant flux approximation.

be found the results of the inteIt is often convenient to have a

flexible, analytical.expression for the temperat~ ~stribution ~ a

radiation wave. Although our results here will be specifically for

Eq. (13), which enibodiestie fundamental assumptions given on page 5,

the method is of far wider applicability. We propose to exhibit a set

of increasingly accurate (and co@icated) approximate solutions to

Eq. (13).

‘lbthis end, we shall convert the differential equation to a pair

of integral equations. A first integration yields

Particularizing with

we find

(28)

(29)

It is a tautology to note that

-16-
IfE. 50

m

T = ~d~.

E E ‘E

Hence, we have

J

(0

dTm

Tin(E)= - — ● de.

E

d~

The scheme for iteration rests on Eqs. (2g)

cessively. There

up (and naming) a

follows tradition

servation, is the

(a) pick a

conditions (19)

of ~/~. with ~.

is a certain amount of freedom

(30)

ad (30)$ applied sucallowable in setting

consistent approximation scheme. Our choice, which

and which has the advantage of automatic energy confollowing:

temperature ctistribution which obeys the boundary

but which is of arbitrary length, i.e., is

as yet unspecified

(b) obtain & (which is proportional to the flux) by

the above approximation in the right-hand side of Eq. (2$1)

(c) obtain Tm(~) by inserting the above apprwcimation

derivative in the right-hand side of Eq. (30)

(d) determine ~obythe condition

a function

inserting

to the

Tm(@ = 1

-17-

(31)
(e) to improve, use the mth root of the above-mentioned expression

fOr Tm(~) as the f~ction ~fermd ‘0 ‘n (a)”

The question is: does this iteration converge on the correct solution? We are in a position to answer this pragmatically, since we can

coqmre it witi the exact solution of Eq. (13)> subJect to the boun~ry

conditions (19). In point of fact, only the first few iterations are

of practical.interest, since a) they turn out to be very good ~d b) the

complexity of the higher iterates is sufficiently great that one may as

well turn to the exact solution itself. We have neither the interest,

the time, nor the erudition to satisfy prurient mathematical curiosity

as to the rate and cause of convergence of the method sketched. It appears to work.

The zeroth approximation. We start by taking for our initisl

perature distribution:

Then if we approximate this step function by a contiguous function

limit is the above, substitute into Eq. (29) and pass to the limit

performing the integration, we find

tem-

(32)

whose

after

-M-
Thus we see that the flux in this order of approximation is constant

behind the head of the wave. Making use of Eq. (30) - step (c) -we

have

[1 (o)

‘rm(k) = 50(E0- E)

Then (step (d))

whence

In passing, we note ~at

[1

(o)

dTm =

-q

g(o) = lo

(34)

(35)

(36)

Equations (32), (34), (3s), and (36) completely specify the zeroth approximation.

-19-
Comments on the approximation scheme. To one coming upon the

constant flux approximation for the first time and viewing the matter

from a mathematical rather than a physical point of vie”w,the construction of the zeroth

recall the form of

(dTm/d~) is indeed

approximation may look somewhat artificial. If we

the exact solution, we note first of SJ-1that

surprisingly constant over a large portion of the

wave. The temperature, on the other h~d~ re~ins q~te close ~ the

driving temperature for a good portion of the length of the wave. It

may also seem inconsistent that

l/m

T‘l)(E) # {[Tm(~)](l)} ●

However, the nature of an appromte solution iS ~at it not be ex~%

i.e., that there be an inconsistency somewhere. The scheme which we

present here has the characteristic that the flux, a derivative qu~-

tity, comes from a somewhat more sophisticated expression for Tm(E)

than the temperature distribution itself. This is in the spirit of a

more balanced approximation as is evidenced by the fact that with this

nomenclature, energy is identically conserved.

The one-halfth approximation. Before returning to our progrmn,

we present an approximation which is sometimes used. We shall call it

the one-halfth approximation, since it turns out that it is intermediate

in accuracy between our zeroth and our first approximation. It also

illustrates another consistent set of energy-preserving approximations.

-20-
Using Eq. (34) and mindful of Eq. (35), we may write

Tin(E)= 1 - E/Eo.

Then

We evaluate f?by taking T as the

‘ ‘l” ‘E ‘l” ‘1

~(1/2)= ~ m

on”

(37)

(38)

mth root of Eq. (37):

- #Eo)l’md5

(39)

Using Eqs. (38) ad (39) in Eq. (18), we find that for energy conservation

Jua .J= ,

0 m

(40)

(41)

We see that the philosophy used here is to take a temperature distribution and use the differential rather than the integral representation

for the flux, obtaining (jOby enforcing energy conservation.

-21-
We have tabulated

Tables II and III, but

higher approximations.

the results of the one-halfth approximation in

we have not carried out the three-halfth or

The first approximation. In accordance with our enunciated scheme,

we take

l/m T(1)(E)= (1- 5/5.) ● (42)

Before we continue with this approximation, we shall find it convenient

to make some transformations. Thus

gives

Another useful transformation comes

Y=l- E/E.●

Then any integral of the form

*

We now see

and this is

Eq. (29) when integratedby parts

(43)*

about by int?mducing the variable

(44)

how energy conservation is guaranteed. For

I

go

dTm

~

o ● 1+

J

~E=t9

g=()= o

the condition, Eq. (18), that energybe conserved.

-22-
e

or

1-~/ (jO

I = toJ’ f(y)dy.

o

Then, making useof Eqs. (29), (42), md (45),

($)1)= -~o[#/m-~,y(~l)/m] .

Integrating once more

[1

(1)

[J

l-~/~.

I

l- E/50y(wl)/m*

Tm(d = -~: Yl/mdy - —

1

0

m+l J o

Then, putting ~ = O to detemine ~o, we

p= V(m + l)(m + 1/2)

o m

.

Using this in Eq. (k6) for y= 1, which

(45)

(M)

(47)

find

corresponds to ~ = O, we obtain

-23-
@ = d-rm .—

d~

I&3

“)=JZEF

Equations (42), (47), (48), and

The second approxtiation.

or

(49)

(49) define the first approximation.

According to our scheme,

We find Eq. (51), the integrand necessary for proceeding

(50)

(51)

with our

second approximation intractable in closed form. However, the quantity

y/(2mtl) has an upper bound of 1/9 for all cases of physical interest.

Hence, a binomial e~ansion of the third factor of Eq. (51) appears attractive; carrying out the e~ansio%we f~d

1

?V =

(2M + l)VV!

T(2) . (m+~/2~’m f ,vy(m+.)/m2+v

W-o

v-1

n (!-J- :), V+o.

As a computational aid, we

v= 1

(52)

notice that

-24-
\

?Hl =

Substituting Eq.

drm

* ‘V* (53)

(51) into Eq. (29) and making use of Eq. (45), we find

This becomes, after some reductions,

(2)

()

dTm

agwhere we have set

v-~ = o

P

m+l =

-----#-”

}

The final integration gives us

[1‘rm(E)

‘2)= ‘~(*r’m r &

,=0

[

. l-( f3+v)

~,-1 + ~~

1

P

,+1

p+, ●

(74)

(55)

(56)

-25-
It is necessary to evaluate this expression and its derivative numerically for ~

(2)

= O to find the values of ~. and 6(2). Actually, the

series converges rapidly even in the least favorable case of m = 4.

*

Equations (51), (54), ~d (56)> togefier ~~

[1 (2)

Tin(o) = 1

and

define the second approximation.

8. Accuracy of the approximations. The results of the

.

approxtiations

are given in Tables II and III.

Except for the one-halfth approximation, the relative erxmrs

(i)

50 /Eo- -i-l 1 are roughly proportional to m as are the relative

rors in ~~i). This illustrates that the scheme furnishes in some

ersense

(i)

an expansion in powers of l/m. At n = O, tie errors in the go ‘S are

(i) 20, 4, and o.~, respectively, ~d in the 80 ‘sj G} 0.8~ and o“%)

respectively.

It is no surprise

errors proportional b

that the one-halfth approximation does not give

m‘3/20 In fact, the error for ~.(1/2) is about

proportional to

O<n<lO, the

absolute value,

m-lo2 (1/2) to m-o” 850 However, for ad that for &o

errors of the one-halfth approximation always lie, in

between those for the zeroth and first.

-x

Five terms give five significant figures

-2GL
The functions T and Tm for n = O, 3, and 10 are given in the figures,

together with such of the approximations as are sufficiently distinct from

them to be drawn. It can be seen that T approaches T

(0) and the flux be.

comes more constant as n grows.

The successive approximations approach T monotonely in the sense that

$i) approaches ~. from below while for values of E sufficiently less than

:, T(i) approaches T from above.

o

-27-
TABLE I

E

(3.00

0.05

0,10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

L

0. 5

0.

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1020

E

o.m

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

::%

0.65

0.70

0. 5

0.L

0.85

0.90

yg:

.

i*O5

1.10

1.15

1.20

n = 0.00 m= 4.00

T

1.000000

0.988029

0.975615

0.962726

0.949327

0.935 76

0.920 i 29

::%%8

0.873043

0.855497

0.836992

0.817411

0.796611

0.774419

0.750611

0.724$x35

~:%@

o:592959

0.547627

0.492578

0.420470

0.306146

0.000000

m

T

1.000000

0.952971

0.905972

0.855036

0.812200

0.765500

0.7189 9

0.672 d 1

0.626655

0.580956

0.535641

0.490 78

0.446 1 38

0.402704

0.359669

:.3&g:49

0:2359 J

0.196927

0.159407

0.123623

0.089937

0.058871

0.031257

0.008784

0.000000

n = 0.50 m= 4.50

T

1.000000

0.989280

0.g78143

0.966559

0.954490

0.941897

0.928731

0.914939

0.go0458

0.885213

0.869119

0.852069

0.83 938

0.81 z567

0.793762

?

0.7 1269

0.7 6759

0.719783

0.689709

c2:275g;;

0.56 z 154

0. 06760

O. ? 16468

0.082325

0.000000

in T

1.000000

0.952656

0.90539

0.8580?0

0.810909

0.763862

0.716976

0.670292

0.623856

0.577718

?

o. 31932

:.48:g7&

0:3973d

0.353676

0.310761

0.268728

0.227723

0.18793o

0.149578

0.11292

z

O.O 85x1

0.0 6g47

0.019414

0.000013

0.000000

go= 1.231173 l?= 0.940689

(Tin)’

-0.94Q689

-0.g40388

-0.939455

-0.937842

-0.935494

-0.932353

-0.928350

-0.923409

-0.917441

-0.91034

-0.90200z

-0.892289

-0.881025

-0.868020

-O.853O34

-0.83566

:$;5 z~:

-0:7657 06

-0.734073

-0.6g6114

-0.64g616

-0.590388

-0.509169

-0.374535

0.000000

m-l T

1.000000

0.964516

0.928616

0.892295

0.855554

0.818387

0.780795

0.742774

0.704322

0.665437

0.626117

0.586359

0.546161

0.505521

0.464437

z

0.42296 “

0.380 2

0.338 ? 99

0.295616

0.252 9

0.208 a 4

0.164231

0.11 516

0.07 : 337

0.0286g4

0.000000

(Tin-l)’

-0.70517

-0.713z36

-$;;;gg

.

-0.7-9072

-0.7i7576

-0.756126

-0.764721

~

-0.7 3361

-0.7 2046

-0.790776

-o. 9 550

-o.n0 368

-0.817230

-0.82616

-0.83026

-0.84?078

-0.853113

-0.862192

do

-0.8 1312

-0.

-0.889:;;

-0.898926

-0.908214

-0.917543

-0.923380

go= 1.200171 ~= 0.g46970

(Tm)‘

-0.g46g70

-0.g46-00

-0.945663

-0.944413

-0.942299

-0.939463

-0.g35840

-0.931355

-0.925921

-0.919439

-0.911 90

-0.902 835

-0.892405

-0.880293

-0.866242

-0.849928

-0.830923

-0.808657

-0.7823

T

%:T%:6

-0.66 038

-0.59 2g64

-0.495199

-0.098801

0.000000

m-l T

1.000000

0.962979

0.925569

0.887768

0.8495 3

0.810 J 3

0.771995

0.732609

0.692821

0.652631

0.612036

0.571034

0.5s624

0.487803

0.44556g

0.40 22

0.359? 5

0.31637 2

0.272477

0.228155

0.18 41o

0.1 J 2 9

:.::::l; t

0:000160

0.000000

(r%’

-0.736532

-0.744301

-0.752110

-0.759957

-0.76784

-0.7 57

z

d

-0.7 3731

-0.791733

LIZ

-o. 99 72

-0.

-&&~

i?

-0:8 2308

-0.820535

-0.848799

-o.8571oo

-0.865438

~

-0.8 3812

-0.8 2222

-0.890669

-0.899152

-o.go 670

z

:::;;4:::

-0.93347

-0.9334i?7

-20-
TABLE I (cont. )

E

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0. 5

0.L

0.85

0.90

0.95

1.00

1.05

1.10

1.15

E

0.00

0.05

0.10

0.15

::Z

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

z

0. 5

0. 0

0.85

0.!30

0.95

1.00

1.05

1.10

1.15

n= 1.00 m = 5.00

T

1.000000

0.990294

0.980197

0.969677

0.958700

0.947223

0.935200

0.92255

0.9092~6

0.895257

0.880396

0.864597

0.847724

0.829611

0.810048

0.788758

0.7653 4

0.739316

::&&5

3

0:585863

0.517209

0.397274

0.000060

n

‘r

1.000000

0.952402

0.904828

0.857306

0.809865

0.762537

0.715355

0.668359

0.621589

0.575093

:.p%

0:437796

0.392982

0.348781

0.305295

0.262645

0.220982

0.180493

0.141421

0.104100

0.069021

0.037011

0.009896

0.000000

n= 1.50 m= 5.50

T

1.000000

0.991132

0.981897

0.972264

0.962196

0.951655

0.940593

0.928955

0.91668

0.9036z6

0.88g887

0.875170

0.859398

0.842398

0.823949

0.803756

0.781424

0.756391

0.727823

0.694396

0.653788

0.601266

:.WC):;

0:000000

m

-c

1.000000

0.952192

:LwOg

0:809004

0.761442

0.714017

0.666761

0.619715

0.572922

0.526430

0.4802s5

0.434580

0.389355

0.344707

0.300735

0.257560

0.215330

0.174238

0.134536

0.og6584

0.060935

0.028608

0.002515

0.000000

60= 1.176525 f?=0.952049

(Tm)‘

-0.952049

-0.951805

-0.951046

-0.94929

-0.947106

-C.945222

-0.941913

-0.937807

-0.932821

-0.926855

-0.919793

-0.911494

-0.901787

-0.8g0461

-0.877249

-0.861806

-0.843674

-0.822222

-0.796542

-0.765244

-0.726021

-c.674529

-0.600616

-0.465297

0.000000

m-1

?

1.000000

0.961736

0.923109

0.884115

0.844754

0.805024

0.764922

:.p;~

0:642378

0.600777

0.558797

0.516437

0.473695

0.4 0569

0.327057

0.343159

0.298873

0.254196

0.209128

0.163667

0.117811

0.071559

0.024909

0.000600

(Tin-l-)’

-:.;;;g9

-017 620

z

z

-0.7 3543

-0.790910

$

-O. 98310

-o. 05743

-0.813208

-0.820706

-0.828236

-0.835799

-0.843393

-c.85102o

-C.858678

-0.866368

-0.8740~

-0.881842

-C.889627

-0.897442

-0.905288

-0.913165

-C.921073

-C.929011

-0.936!?80

-0.941220

go= 1.157887 ~= 0.956241

(Tm)‘

-0.956241

-0.956018

-0.955324

-0.954118

-o.952355

-0.g49981

-0.946936

-0.943151

-0.938545

-0.933020

-0.926462

-0.918731

-0.909658

-o.899Q28

-0.886567

-0.871920

-o.8546o3

-0.833938

-0.808g24

-0.777979

-0.738350

-o.68444g

-0.601270

-o.389493

0.000000

m-1 T

1.000000

0.960711

0.921080

0.881106

0.840788

0.800125

0.759114

:.pi”;

0:633994

0.591570

0.548802

0.505679

0.462199

0.418360

00374162

0.329603

0.284681

0.239396

0.193746

0.147729

0.101345

0.054591

0.007467

0.000000

(P-1)‘

-0.782379

-0.789195

-0.796039

-0.802912

-0.809813

-0.816742

-0.823700

-0.830686

-0.837699

-0.844 41

-0.851~10

-0.858907

-0.866031

-0.8 3183

-0.8z0362

-0.887569

-0.894802

-0.902063

-0.909351

-0.916665

-0.924006

-0.931374

-0.938769

-0.946189

-0.947362

-29-
TABLE I (cent. )

c

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0. 5

o.z0

0.85

0.5U

0.95

1.00

1.05

1.10

E

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

1

0. 5

0. 0

0.85

0.90

0.95

I.00

1.05

1.10

n = 2.00 m = 6.OO

T

1.000000

0.991838

0.983329

0.974444

0.965147

0.955401

0.94515

0.93436x

0.922956

0.910860

0.897983

0.884213

0.86g411

0.853401

:.:E3~9~:

0:795x07

::;l; ?g

0:669590

0.614746

0.527001

0.600000

m

T

1.000000

0.952015

0.$)04052

0.856132

0.808280

0.760524

0.712893

0.665419

0.618140

0.571097

0.524334

0.477905

2

0.4 1868

0.3 6294

0.341263

0.296874

0.253243

0.210521

0.168900

0.128642

0.090126

0.053973

0.021422

0.000000

n = 2.50 m= 6.50

T

1,000000

0.992439

0.g84551

0.976306

0.96 671

0.9J 608

0.949071

0.939007

0.928356

0.917040

0.904971

0.892035

0.8780g4

::%?2;

0.828147

0.807 01

0.784x4

0.75 9

?8

J

o.72 95

0.683773

0.626596

0.525320

0.000000

in T

1.000000

0.951865

0.903750

0.855675

0.807664

0.759742

0.711935

0.664276

0.616799

::%::

0.475864

0.429552

0.38 675

0.33d313

0.293560

::%?:3

1

i

0.16429

0.123534

0.084514

0.047909

0.015232

0.000000

E.= 1.142817 8=0.959760

(A ‘

-0.959760

-0.959555

-0.g58915

-0.957803

-0.g56174

-0.953979

-0.951160

-0.947650

-0.943369

-0.938225

-0.932105

-0.92ti72

-0.916356

-0.g06344

-0.8g4560

-o.88063g

-o.86407g

-0.844167

-0.819821

-0.789284

-0.749356

-0.693062

-0.598510

m-1

T

1.000000

0.959850

Zz

0.91 3 9

0.87 5 5

0.837468

0.796026

0.754258

0.712163

0.669740

0.626986

0.583902

0.540486

0.496737

0.452653

0.408233

2Z

0.36 4 6

0.31 3 2

0.272948

0.227174

0.181058

i

0.1 4600

0.0 7797

0.040650

(Tin-=)’

/i

-0. 99800

-o. 06209

-0.812643

-0.819102

-0.825586

-0.83tig4

z

-0.8 8626

-0.8 5183

-0.851764

-0.858370

-0.864ggg

-0.871653

-0.8 8330

-0.8J5032

-0.891758

-0.898507

-0.905280

-0.912076

-0.918896

-0.925739

-0.932606

-0.939496

-0.946409

0booboo o.Ooooilo -0.s352348

gO= 1.130378 G= 0.962756

(Tm)‘

-0.962756

-0.962565

-0.961972

-0.960940

-0.959427

-0.95 386

-0.95?761

-0.951488

-0.947491

-0.94269

-0.9369T3

-0.930148

-0.9221

-0.9126%

-0.901494

-0.88826

-0.8723L

-0.853182

-0.82$3496

-0.799403

-0.759263

-0.700561

-0.591357

0.000000

m-1 T

1.000000

0.959117

0.917931

0.876442

0.834647

0.792547

0.750139

0.70 424

0.66x399

0.621o64

0.577418

Jd

0.5 3459

0.

!

‘I87

0.44 600

0.399698

0.3544 9

0.3089i2

0.263087

0.216912

0.170417

0.123600

0.076459

0.028996

0.000000

(T%’

-0.81469

-0.820623

~

-0.82649

-0.832 6

-0.8389?5

-0.845076

-0.851227

-0.857401

-$:;3 ~~

J

&

-0:8 6048

-0.8 2306

-0.88886

-0.8g4386

-0.s01207

-0.907549

-0.913912

-0.920296

-0.926700

-0.933125

-0.9 9570

-0.926036

-0.952523

-0.956474

-30-
TABLE I (cont. )

E

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

~

0. 5

0. 0

0.85

0.90

0.95

1.00

1.05

1.10

E

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0. 5

8

%8;

0.90

0.95

1.00

1.05

1.10

n= 3.00 m= 7.00 Eo= 1.119935 e= 0.965336

T

1.000000

0.992958

0.985606

0.977916

0.969854

0.961385

0.952463

0.g43038

0.g33048

0.922420

:.g;!x:g

o:8856g4

0.871362

0.85563

0.83819z

0.8185g4

::;%8

0.737851

0.696590

0.637040

0.517073

0.000000

m

T

1.000000

0.951 36

0.903I90

0.855282

0.807134

0.759067

::W

0.615641

0.568198

0.521003

0.474101

0.427549

:.WO:

0:290686

0.246309

$%%J

0:079587

0.0425 6

0.009812

0.000000

(Tm)‘

-0.965336

-0.965159

-0.964606

-0.963643

-0.962231

-0.960323

-0.957868

-0.954803

-0.951054

-0.946534

-0.941137

-0.934730

-0.927151

-0.918188

-0.90 567

-0.89I916

-0.879713

-0.861191

-0.838150

-0.808523

-0.768243

-0.707092

-0.577616

0.000000

m-1

T

1.000000

0.g58486

0.916685

0.874597

0.832221

0.789556

0.746601

0.703355

0.659818

0.615987

0.571862

0.527443

0.482728

0.437716

0.392407

0.346 99

0.300z92

0.254685

0.208177

0.161366

0.114252

0.066835

0.019112

0.000000

(Tin-l)‘

-0.827431

-0.833146

4*(33888(-J

-0.844633

-0.850405

-0.856197

-0.862007

-0.867836

-0.873684

~

-0.8 9551

-0.8 5436

-0.8g1340

-0.897263

-0.903205

-0.909165

-0.915143

-0.921140

-0.927155

-0.933189

-0.939240

-0.945310

-0.951398

-0.957505

-0.959944

n= 3.50 m= 7.50 ~0= 1.111043 E= 0.967582

T

1.000000

0.993410

0.986526

0.979320

::%; z:;

0.955432

0.946569

0.g37164

0.927144

0.916422

0.904888

0.892403

0.878 88

0.863L 7

0.84 136

0.82J314

0.806655

0.781058

0.749581

0.708241

0.646255

0.497012

0.000000

m

T

1.000000

0.951624

0.903264

0.854g40

0.8066 2

0.7584i1

0.710392

0.662433

0.614633

0.567029

0.51965

0.47256x

0.425801

0.379430

0.333523

0.288169

0.243481

0.199600

0.156720

0.115117

0.075224

;.:;03;:;

0:000000

(Tm)‘

-0.967582

-0.967416

-0.966899

-0.965997

-0.964672

-0.962882

-0.960576

-0.95 693

-0.95x164

-0.949903

-0.944807

-0.938747

-0.931564

-0.923050

-0.912931

-0.900837

-0.886240

-0.868358

-0.845940

-0.816794

-0.7’i’6433

-0.712766

-0.551470

0.000000

m-l T

1.000000

0.957936

0.915601

0.872993

0.830113

0.786959

0.743530

0.699825

0.655844

0.611586

0.567050

0.522235

0.477140

0.431765

0.386108

0.340169

0.293947

ywl;

0:153575

0.106213

0.058563

0.010626

0.000000

(Tin-l)’

-0.83851

-0.843d 9

-0.849424

-0.854876

-0.860344

-0.865830

-0.871332

-0.876852

-0.882388

-0.887941

-0.893510

-0.899096

-0.904699

-0.910318

-0.915954

-0.921606

-0.927274

-o.932959

-0.938661

-0.944378

-0.950112

-0.955862

-0.961628

-0.962904

-31-
TABLE I (cont. )

E

0.00

0.05

0.10

0015

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

z

o. 5

0. 0

0.85

::%

1.00

1.05

1.10

E

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

1

0. 5

%8;

0.90

0.95

1.00

1.05

n = 4.00 m= 8.OO

T

1.000000

0.993808

0.987336

0.980557

0.973441

0.965953

0.958052

0.949688

0.940803

0.931327

0.921172

0.910232

0.898368

0.885403

0.871102

0.855137

0.837043

0.816114

::%%%

0.718887

0.654377

0.441361

0.000000

m

T

1.000000

0.951525

0.903066

0.854640

0.806266

:.w;;~

::::;:~;

0:566000

0.518474

0.471211

0.424262

0.377685

i

0.3 1551

:.;4;;::

0:196791

0.153574

0.111605

0.071332

0.033622

0.001440

0.000000

n= 4.50 m= 8.50

T

1.000000

0.994160

0.988054

0.981654

0.974933

0.9678 5

0.9603z1

0.952463

0.944044

0.935055

0.925412

0.915007

0.90370

0.89133I

z?

0.8 76 4

0.8 23 1

0.844923

0.824683

0.800439

0.770019

M%

0.000000

m

T

1.000000

:. 95;:;;

0:854373

0.805907

0.757509

0.70$)202

0.661011

0.612963

0.565089

0.517424

0.470011

0.422896

0.376136

0.329799

$

0.28 969

0.23 752

0.194286

0.150 62

0.108T59

0.067836

0.029824

0.000000

(0 = 1.103380 8= 0.969555

( Tm)‘

-0.969555

-0.969399

-0.968913

-0.968064

-0.966817

-0.965131

-0.962956

-0.960236

-0.956902

-0.952872

-0.948045

-0.942298

-0.935472

-0.927364

-0.917704

-0.906122

-0.892089

-0.874809

-0.852995

-0.824335

-0.783941

-o. 17667

-o.?86803

m-l T

1.000000

0.95 453

0.91x649

0.871586

0.828263

0.784681

0.740838

0.696733

0.652365

0.607 35

0.562z41

0.517682

0.472258

0.426568

0.380611

H

0.3 4 87

0.2 7 94

0.241132

0.194101

0.146799

0.099226

0.051380

0.003263

0.006000

(Tm-h‘

-0.848361

-0.85 509

-0.85i673

-0.863852

-0.869046

-0.874255

-0.8 9479

-0.8i4719

-0.889973

-0.895242

-0.900526

-0.905825

-0.91113

-o.g16462

-0.921811

-0.927170

-0.932543

-0.937930

-0.943333

-0.948750

-0.954181

-0.95962

-0.96508Z

-0.965457

EO= 1.096707 8= 0.971302

(?m)’

-0.971302

-0.971155

-0.970696

-0.969895

-0.968717

-0.967123

-0.965066

-0.962490

-0.959331

-0.955508

-0.950925

y; ;Iy

3

-0:931219

-0.921 9

-0.910r70

-0.897361

-0.88064g

-0.85g416

-0.831245

-0.790857

-0.721853

0.000000

m-1 T

1.000000

0.957026

0.913806

0.870340

0.826628

0.782668

0.738459

0.694002

0.649295

0.604337

0.559129

0.513669

0.467957

0.421992

0.375773

0.329300

0.2825 2

82

0.2 5z8

0.1

0.140222

0.0930 7

0.045085

0.000000

(Tin-b’

-0.857031

-0.861935

-0.866852

-0.871783

-0.8 6727

-0.881686

-0.886657

-0.891642

-0.8g6641

y090g;3

-0:911712

1

-0.916 71

-0.921 36

-0.92691

-0.9320d

-0.937114

-0.942233

-0.947365

-0.952510

-0.957668

-0.g62840

-0.967683

-32-
TABLE I (cont. )

E

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0. 5

~

&8;

0.90

0.95

1.00

1.05

E

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0. 5

0.$0

0.85

0.90

0.95

1.00

1.05

n= 5.00 m = 9.00

T

1.000000

0.994475

0.988695

0.982634

0.976266

0.969556

0.962465

0.g54g48

(y#

0:929218

0.919300

0.908513

0.896681

0.883573

0.868861

0.8520 5

0.8324z4

0.8088 7

0.7789&5

::22%;:

0.000000

m

T

1.000000

0.951359

0.902733

0.854136

0.805586

0.757102

0.708704

0.660415

0.612263

0.5642 6

0.5164J8

0.468939

0.421676

0.374751

0.328231

0.282198

0.236754

0.192037

0.148233

0.105623

0.06467g

0.026390

0.000000

n= 6.OO m= 10.00

T

1.000000

0.995012

g.$#fz

0:978547

0.972469

0.966038

0.959212

0.951938

0.g44152

0.935775

0.926705

0.916816

0.905938

0.8g3843

0.880208

0.864561

0.846162

0.823 52

0.794195

0.753751

0.677664

0.000000

m

T

1.000000

0.951226

0.902465

0.853731

0.805040

0.756407

0.707853

0.659398

:.dg:7

o~514881

0.4671o8

0.41g588

0.372380

0.325544

0.279159

0.233322

0.188163

0.143868

0.100716

0.059194

0.020424

0.000000

EO= 1.090844 E= 0.972859

(Tm)‘

-0.972859

-0.972720

-0.972286

-0.971527

-0.970411

-0.968900

-0.966948

-0.964503

-0.961501

-0.957866

-0.95 502

-0.948291

-0.g42084

-0.934685

-0.925831

-0.915158

-0.902138

-0.885962

-0.865286

-0.837602

-0.797254

-0.725369

0.000000

m-1 T

1.000000

0.956645

0.913055

0.869231

::%:;;

0.736342

0.691572

0.646564

H%;:

0.510105

O.464I38

0.417931

:.;:;48:

0:277z57

0.230679

0.183257

2

0.1 5591

0.O 7680

0.039522

0.000000

(Tin-l)’

-0.864 64

-0.869I44

-0.874136

-0.8 8841

-0.8i 558

-0.882288

-0.893029

-0.897783

-0.902549

-0.907328

-0.912118

-0.916921

-0.921736

-0.926562

-0.931401

-0.936252

-0.941115

-0.945990

-0.950877

-0.9557 6

-0.9606A7

-0.965609

-0.969640

E. = 1.081022 e= 0.975517

(-Tm)‘

-0.975517

-0.975391

-0.974g98

-0.974312

-0.973302

-0.971933

-0.970163

-0.967943

-0.965213

-0.961902

-0.957920

-0.953156

-0.g47466

-0.g40662

-0.932492

-0.922600

-0.g10463

-0.895271

-0.8 5642

-0.8t8g14

-0.808726

-0.730469

0.000000

-33-

m-1

T

1.000000

0.955995

0.911775

0.867339

0.822689

0.777822

21

0.7 38

%:479:;

0.596182

0.550227

0.504052

0.457658

0.411043

0.364208

0.317151

0.269873

0.222373

0.174650

0.126703

0.078533

0.030139

0.000000

(Tin-l)’

-0.8 7965

-0.882252

-0.886550

-0.890858

-0.895176

-0.899505

3

-0.90 843

-0.90 192

-0.912551

-0.916920

-0.921299

-0.925688

-0.930087

-0.9344g6

-0.938916

-0.94.3345

-0.g47784

-0.952234

-0.956693

-0.961162

-0.965641

-0.970130

-0.972920
TABLE I (cont. )

E

0.00

0.05

0.10

0.15

0.20

0.25

0.30

?0

o. 5

0.

0.45

0.50

0.55

0.60

0.65

0.70

~

0. 5

0. 0

0.85

::%

1.00

1.05

5

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0955

0.60

0.65

0.70

00 5

0.80

0.85

0.90

0.95

1.00

1.05

xl= 7.00 m=ll.oo

T

1.000000

0.995454

0.990692

0.985692

0.980428

0.974873

0.9689 0

0.9627?0

0.956071

0.948923

0.941220

0.932866

0.923738

0.913673

::3%;:;

0.875099

:W&

0:76 793

0.68T350

0.000000

m

T

1.000000

0.951117

0.902246

0.853399

0.804s90

0.755836

1!

0.70 154

0.65 562

0.610084

0.561744

00513571

0.465599

0.417867

0.370423

0.323325

0.27664s

i

062 0477

0.1 4946

0.140233

0.0g6614

0.054592

0.015419

n= 8.OO m = 12.00

T

1.000000

0.995824

0.991447

0.986849

0.982006

0.976891

0.971470

0.965705

0.959549

0.952942

0.g45814

0.938071

0.929597

0.920232

z

0.909 61

?

0.89 75

0.88 112

~

0.86 31

0.847 26

0.820525

x

007 9943

0.6 7580

0.OOOOOO

m

T

1.000000

0.951025

0.902062

0.85312o

0.804215

0.755359

0.706569

0.6s7863

0.609261

0.560 8

0.512Xzl6

0.464335

0.416424

0.368 82

0.321?61

0.2745 1

0.2280i1

0.182231

0.137157

0.093133

0.050670

0.011166

O.ocoooo

E. = 1.073118 e= 0.977700

(-Tm)’

-0.977700

-0.977585

-0.977227

-0.976601

-0.975679

-0.974428

-0.972809

-0.970775

-0.968273

~:.;:3;3:

-0:9571 z 3

-0.951931

-0.945636

-o.9~8053

-0.928837

-0.917476

-0.903162

-0.884493

-0.858690

-0.818741

-0.732950

0.000000

m-l

‘T

1.000000

0.955460

0.910723

0.865787

0.820652

0.775318

0.729784

o.684Q5o

0.638116

0.591981

0.545644

0.499106

0.452365

0.40 422

0.353276

0.310926

:.;:;~7:

::;:;$;

0:071110

0.022531

0.000000

(Tin-b’

-0.888818

-0.892772

-0.896735

-0.900707

-0.904687

-0.908676

-0.912673

::%J9

. 2

-o:g28@18

-0.932 88

-0.93d 36

-0.940893

-0.944959

-0.;4~32

-0.953115

-0.957205

-o.9613o5

-0.965412

-0.969528

-0.973652

-0.975562

E. = 1.066619 ~= 0.979526

(?m)’

-0.979526

-0.979421

-C.979092

-0.978516

-0.977667

-0.976515

-0.97502

d

:::%?:3

-0.96802z

-0.g64639

-C.960572

-C.955696

&@~;

-0:934141

-0.923466

-0.

r

9939

-o. 92151

-0.867232

-0.827578

-0.732435

0.000000

)

m-1

T

1.000000

0.955013

?

0.90 43

0.86 489

0.818950

0.773227

0.727319

0.681225

!J

0.6 4946

0. 848o

0.541828

0.494989

0.447g62

0.400748

0.353346

0.305756

0.257977

o.2100c9

0.161852

0.113504

0.064967

:.:()::

9

(P-l)’

~$8#7g;

-O:$W5243

-:.W&:

-0:916314

-0.920019

-0.923731

-0.927451

-0.931178

-0.934912

-0.938654

-0.942402

-0.946158

-0.949921

-0.953692

-0.957470

-0.9612 4

d

?

:::;6 :42

-0.972652

-0.976465

-0.977735

-34
TABLE I (cont. )

E

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

E

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0. 5

0.80

0,85

0.90

0.95

1.00

1.05

n = 9.00 m = 13.00

‘r

1.000000

0.996139

0.992090

:::%$;

0.978609

:::ZZ:$!

0.962516

0.9563 5

0.9497I2

0.942528

%82;s

0.916055

0.904887

0.891910

0.876388

:.;;700;

0:790783

0.686460

0.000000

In

T

1.000000

0.950948

0.901906

0.852884

0.803896

0.754953

0.706072

0.657269

0.608563

0.559975

0,511531

0.463260

0.415197

0.367384

0.319873

0.27z728

0.226035

0.179909

0.134521

0.090141

0.047288

0.007516

0.000000

n = 10.00 m = 14.00

T

1.000000

0.996409

0.992642

0.988681

0.984505

0.980089

0.975405

0.970415

0.965078

0.959342

0.953139

0.g46386

0.938974

0.930755

0.921527

0.910998

:.EM~3

0:86544T

0.840318

0.800463

0.678353

0.000000

m

T

1.000000

0.950881

0.901772

0.852681

0.803622

0.754605

0.705646

0.656759

0.607962

0.559276

0.510725

0.462336

0.414141

0.366181

0.318504

0.271172

0.224267

;:;;;F6

0.087540

ywl;~:

0:000000

E.= 1.061182 E= 00981075

(Tm) ‘

-0.981075

-0.g80g78

-0.g80674

-00980141

-0.979355

-0.978287

-0.976904

-0.975164

-0.973018

-0.970406

-0.967253

-0.g63463

-0.958913

-0.953437

-0.g46810

-0.938708

-0.928642

-0.915823

-0.8g8844

-0.874766

-0.835448

-0.727869

0.000000

m-1

T

1.000000

0.g54634

0.90909

0.86338 8

0.817508

0.771456

0.725231

0.678833

0.632262

0.585518

0.538600

0.491508

0.444242

0.396801

0.349185

0.301394

%%Z$8T

0.156966

0.1O8471

0.059799

0.010949

0.000000

(’rm-l)’

-0.905608

-0.909028

-0.912455

-0.915888

-0.919328

-0.922773

-0.926225

-0.929684

-0.933149

-0.936620

-0.940097

-0.943580

-0.947070

-0.950566

-0.954069

-0.957577

-0.961092

-0.g64613

-0.g68140

-0.971674

-0.975213

-0.978759

-0.979553

E.= 1.056566 g = 0.982407

(Tm)‘

-0.982407

-0.982317

-0.982033

-0.981537

-0.980806

-0.979811

-0.978521

-0.976899

-0.974896

-0.972456

-0.969507

-0.965960

-0.961695

-0.956554

-0.950321

-0.942681

-0.’333159

-0.920982

-0.904745

-0.881466

-0.842511

-0.716406

0.000000

m-1

T

1.000000

0.954308

0.g08456

0.862443

0.816270

0.769935

0.723439

0.676781

0.629961

0.582979

0.535835

0.488527

0.441057

0.393423

0.345626

0.297665

0.249539

0.201249

o•152795

0.104175

0.055391

0.006441

0.000000

(Tin-l)’

-0.912235

-0.915439

-0.g18647

-0.921862

-0.925082

-0.928308

-0.931539

-0.934775

-0.938017

-0.941265

-0.944518

-0.947776

-0.951041

-0.954310

-0.957585

-0.960865

-0.g64151

-0.g67442

-0.970739

-0.974041

-0.977349

-0.980662

-0.981097

-35-
i!

H

Nwmmmm mmmm

‘~ ‘o ‘o ‘~ ‘o ‘~ ‘~ ‘o ‘o ‘o

l-ii+ l-l I-I t-it-l

. . . . ● . . . . .

-ao

-2

o

U“

I

II

-a

w

-36-
.

t-l

w

A

.

w

90

la

‘o

NOJCM

‘o j

rt l-1

. . .

‘o

NOJCU

‘~ ‘:

l-l

m

n

0 0

al

d

‘o

m

.

(n

Cu

l-i

‘o

.

i-i

t-l

II

.37-
IIIIIIIIIIIIIIIIIIIIII1IIIIIIIII

.1

IIII

-1 

/“

I

----- ---—,

I

,,

?,

/’

/’

/’

/’/; (0

/’

—o“

/

1’

; —2 i/ * i —o

1’;

m

I —d i N

—o

.: —z oII q G o d s = G o

1- -i-i- -1 -b

c

11

II I 1!

11,

II

b

-38-
--

z=

-f

EEE

o~k~

c

11

1::

1,

—

q

0

—

q

0

—

m

0

— 6

— z

—-

0

~ 0“

E

1-

-39”
).

1- ----- --- J------------------ix

t./

I

/

. II I I /’ 1I

/’

[/

II

/’

II

/’ I

I

1! I

I

II

1’

II 1’ I 1’ I

:1

11

1,

1;

II

1,

1;

;;

II

1!I

1!

1’

1;

1’

1’

11’

I

!IIIIIIIII

g=

I-K -i

c

It m I I I I

II

I

I i 3i —1

0-l

1

II

II

I

II

II

I

o

g —- 3

-40-
=i-

-41-
I

-----

IIIIIIII1IIIIIIIIIIIIIIIIIIIIIIIIII[IIIIIIIIIIIIIIIIIIIIIIIII

------- ------------ . =0 — 3—ks— 0 —

u) 8

—

0

—

IQ~

—

CN0

—

0

In

-42-
:

A

0

/—

/

/

/

/

/ ~

/

/

1

/

/

1IfJ

2

[ I I I I I I I I I

0 IQ

o:

—. 0 0

-43-