U. S. AIR FORCE

PROJECT RAND RESEARCH MEMORANDUM

A TJJ3LE OF TH:E: PL:;,.NC!: P.'-..Dli.TIO!" FU1iC':'ICC

Assigned to ------------

This is a working paper. It may be expanded, modified, or with­drawn at any time. The views, conclusions, and recommendations expressed herein do not necessarily reflect the official views or policies of the United States Air Force.

------7MR.!l n De.~ 1700 MAIN ST. • SANTA MONICA • CALifOINI.A-----

A TABlE OF THE PLANCK RADIATION FUNCTION AND ITS INTEGRAL

RF-l'7L3

ABSTRACT , .. ,, ··~··"'<·"-~·--·-~·-

RY-l 71~ i '7-'?-~6 - i~ -

1 X X/( f'c ·- 1 \

a:>d its n"!'!"l.aliz,ed inte~ral~ (Jr./r/·') Jx xi/(ex- J) dx. t<" f"ye s:i::nificant ',

figures, for x:. 0 (0.02) 10. It also gives fjyp.-figure val,l"''? fo:::- the P1a,,~)-·

RN-1 '7L:3 '7-?-~6

- 1-

ThF> rl'ite at whi t:'h a " black body" (nerf~ct raciator) emits radian+ ene;-,r:rY

at frec:pF>ncief' betw~?>en v anri 111+ dV was deduced b;r Plane¥- to be

per unit area, where

h = 6.6?5? X 10-27 ere:- sec (Planck'f' constant,),

c = 2.9q'79 X 1010 em/sec (velocity of lii3ht).

k = 1.)80/c X 10-16 ergf>K (Boltzmann's constant).

anc T i.s the ab~r:lute temn<e:rature of the black body. ThP t-otal rad'i ation rate

far al1 frequo<>ncies between 0 and v is then given by the int.er."ral

'?-;;(! IV v3ctv 2-:;(l(r{ (X ~,

----l ---·---· = J 2-'...9:.:_'

2 hvikT 1 h\2 X J c e - e -0 0

whe,..e the dimensionless va,..iable :x ~oualP hV/kT. The tctaJ. radiation rate,

integrated over all frequencies, is

where

0 = =

3. X C1Y

ex- 1

= 2"" ~(J(J:l4 = ~ ?

] 5 h c-

6A -s 1 ? o 4 s. _B'7 x 10 - erg em sec K

rr>lt o_l '

The fra 0 tion of the total radiation rate which is radiated at freouend.es S"ch

that hv/kT is between 0 and x is

.c;3,dx._ ex_ 1

R~-'-l7i.~

'7-?-t::.A 2

In the d~.scuesion to foJ.low, this last exp~eseion w:1 11 be ca1Jer1 the normAJ i?.ed

Plan:-1.< integra 1 •

The Planck integral cannot be reducl!'d to elementa:r:r functione, sc a table

of tb:i.c.: inte;,;~al is a. usef•_:l tool for making many typt:>s of' radiat:ion cAl C1J.la.ticnc.:.

At leact three sn::-h tables have been pubJishec preYiousl:' (seE' LNm~ all~ Rla:nc1l(1)

anr1 th"' !'ef'erences listecl therein), b1Jt alJ of then1 are in dimen::-ion;.J for:r arx3

are bac.€'r\ on olcier Ya1ue:: of Plan~k's constant which diff"lr in t.h"' fourlh fi.P''.J!"P

from the modern value quoted ab,ve. In addit::on, thev tab,,late the integ"cl fo-r

freo111?n~;r " tan" (x !:::. 0. 7) • ;-; ,·

f ~

( '3 1 t:: 7 Q 1 t:: --~-<2! .. L;

o.1 ~:.3::;::;c ~-. X y

= + ..... ...._... ........ + l~ - :-r-v 3 50LO e «.

1 .. ) 'IT -

\ .. )

At x = 2C VI'RS obtained from the serie::>

11) l~ i~ r ( ~ X OX -<·. = 0.153oqo l x-l. :f. TI' e - l

+

( ~3 ') c) 2

~.

'iv- 6-:-r 6 ) -:?.x 'ix 6v: t.. -'?~I i \,/ -· .. I + + + + 16 e + + + + e +

h 8 3 9 ')7 -81 .. -! i

-'

RH-2.71:':\ 7-?-t::.l-,

- 3 -

The samE' series was used to check several other points; aa;reemen+. tC' f:ive

sig:n.:ficant figures was found.

A further check was Provided by the norma1izatjcn concition:

ro ':< Lro =~~xl lt:; ...z.:<ci~£._. + ]:..: = 1 4 4 • X 1 1T e - rr

0

Sinc-e th:} s relation was satisfied to 0.000()1 an0 the prcces~ of inteP"ratj "'n wa::

c<v;-iulative (eac!! value der-endinr; unon the values at the prev:i0us pcil!+s). this

im'l"llies that ever;· value is accurate to at least 0.00001 (i;:rnor5np· thP nossi!'-iJit:r

of COC":nensatin"' errors). Of course for those values of the norma1j -zer1 int.f'~.,..aJ

After tl-i s table was comp'1ferl and in

t"x was issued wticr' incl:Y'es eight-fif'1'ure ta'b1Pc:: l'f' )

u('

x3/(ex- 1) dx f0.,.. x = 0.1. (0.0:!) }n (0.1) ?S (J)

1~ I.o-v:~n, A, N. anrl G. FJ.anch, tt Table<:: of Planr}.rts Radiation and P..t1n,.._tj0n~tt4t -~T-·~-.!l:_,t_. __ ?:!'...;:_. __ ,"~~~Tfi~...,., \lo1. 3CJfj Feb:rua..,.y_~ 194.0, p. 70. jsf'I..,r' HS A. \rJ.P.A"• Table, N€'"' York~ 101;1)

R~I-171,':<

7-2-1:;6 - 4 -

Ph~+.o"l

(Also

2. 12'-'a:-P .. R. L.. li'. L. "'(iller, W. E. Pearson and J. F. Peo...,l'-'"', J;'at;!_""~"' .. PJ' £la'::}. J3_1)..c!.:LJ:'.§:"l~.?:+.j:.2.2E~--:tte Tral}~""i :§.iS'n F§St ?.::_t~:r Ra¢.i~l:L9n._Th:-c~ li_.q_:~r_V?Jl.1.!:· NI\\TC?:D P.:-port 3171~ November 15, lOt:;c: .•

RH-1743 7-2-~6

x3 x3 - 5 -15 ( Ji

-- dx X -1 X -1 X e e

0 0 x1o-4 0 '7

.02 3.9601 4.0757 xlO .. ,

.oL;. 1.5682 xlo-3 3.2361 x10-~1

.06 3.h?3l 1 1.08hO x10-:;

.oB 6.lL n~ 2.5~01 t .10 9.5oE3 4.9431

.12 1.3553 x1o-2 8.4770 I ,... _,

.14 1.8260 1.3359 x10 ...,.

.16 2.3607 1.9790

.18 2.9571 2.7964 ' ' t .20 3.6134 \} ). fJ"ibt,

!

.22 4 .. 3270 I 5 r-, ,..,?" i •': c. ':'

.24 5.n964 ' 6.4no

1 ! ')~ 5.?192 8.1726 o<-t' w

.26 6.7936 1.0127 xl'"'-3 .. \_1

.30 7. 7174 1.2362 t

.32 8.6888 1.hhf:7

.34 9. 7059 _, 1.7719 o3b 1.0767 xlO -'- 2.0E:70 .38 1.1870 2.u355 I

t I .40 1 • .3013 2. flB6 If'

.u2 1.~194 3.23'!)1

.h4 1.~412 3.h)l32

.u,6 1.6665 4.15?1

.L;.b 1. 7951 h. 7201

.so 1.)'269 If 5.2932 "

.52 2.0616 j 5.9073 c:'4 2.1992 6.5b}3 ·--.56 2.3395 7.2622

.58 2.l.J.t22 8.CC!.J.6

.60 2.6274 8.7914 'It

n"' 2. 77L~ 7 9.6232 ~ • ,_t_

-2 .64 2.9242 l.05C1 x•n .;..I._;

.ht· 3.0755 1 - l ')4 ol<-:'-

.66 3.2286 1.2395 • 70 3.3835 1.3413 y

.72 3.5398 1.1~.1~5:·

l • '(4 3.69'75 1.559h .76 3.13565 1.6757 70 4.0166 1. 7970 • v

.Bo 4.1778 1.9231

PJ-1-1743 7-2-56 - b -

x3 15 X x3 ;,4 r -- dx

X -1 X -l X e e 0

.82 1.+-3398 x10 -1 2.05L3 x10-2

.e4 4.5026 I 2.1904 I I

.t6 4.66tD I 2.3316 l

.r.ts 4.8301 I 2.4 7'19 ~ I

.90 4.9945 \}' 2.6286

.92 5.1:>93 2. 7855 04 6/ 5.3243 2 ()hhQ 0/ .._J/

.. 96 5.h895 3.1J..35

.90 5.o;,Lo 3.2E:;,l 1~QO 5.b195 ¥ 3.4615

1.02 5.984 7 3.6L>.J6 1.01.1 6.1)(94 3.8)0~ 1.06 6.3138 4.0223 1 ()P o-v 6.L, 7Tf 4 01(.)1 •'--""...,. 1.10 6.6412 'it 4 4"-~ • .O:.l) t

1.12 6.80't0 h.62ttl 1.14 6.9662 4.8401-4 1.16 7.1276 5.05'74 1.18 7.2t02 5.2794 1.20 7 .~4 79

' 5.5064 'f

1.22 7 .tJJCt- 5o 7)62 1.24 7.7644 5.971+9 1.26 7. 92lC! 6.2164 1.2b 8.076!.~ 6.[[628 1 • .30 t.230b 6.7l3Y

1.32 0.3836 6.96?7 1.34 8.53~2 7.2JC3 1.36 8 (:;4 7 .L:955 .6Jj 1 ')q 8. 8342 7. '(652 o..;O

1.l.Jo 8.9614 \ 6.0396

1.42 9.12'(1 6.3185 1.44 9.2712 8.ocn8 1.46 9.1.!.137 8.h895 1.48 9.5545 9.1Ea6 1.50 9.6936 'i 9.t;78o v

L.52 9.8309 9.7766 ·< -1 1.54 9.9664 ~ l.OC·83 x10 1.56 1.0100 x10° 1.0392 1.58 1.C232 1.0705 1.to 1.0362 '¥ 1.1022 v

RH-1743 7-2-S6 - 7 -

x3 X

X3 1.5 s -- dx X ~ X X e -1 e -1 0

x10° _,

1.62 1.0490 1.1344 xlO ... 1.64 1.0616 1.16tt ! . 1.66 1.0740 1.1997 i

t 1,6o 1.0662

i 1.2330 t 1. 70 1,09t1 .'II 1.2666

1. 72 1.1099 1,YC6 l. 74 1.1215 1.33.50 1.76 1.1328 1.3697 1.78 1.1440 I 1.u:~L~s 1.8o 1.1549 J· 1 hL'V' • .\..·C:. ~.

1.82 1.1656 1,L759 1.84 l.l ?62 1 • .5114 1.86 1.1864 1,51J t~ 1.8b 1.1965 1 • .5850 1.90 1.2063 .~· 1.6221

1.92 1.Cl59 1.6594 1.94 1.2253 1.6970 1.96 1.2345 1.7)48

I 1.Sl8 1.243l.l 1.7730 I I

2.00 1.2521 1.en.4 I

f

2.02 1.2606 1.5501 2,04 1.2689 1,6891 2.06 1.27t9 1.)'283 2.oe 1,28lr7 1.9677 } 2.10 1.292.3 2.0C74

2.12 1.2997 2.0473 2.1)-l. 1.)068 2.C87I-! 2.16 1.3137 2.2178 2.1c 1.3204 2.1654 2.20 1.3260 2.2C9l

2.22 1.3331 2.2.501 2.24 1.3391 2.2912 2.26 1.3449 2.]326 2.28 1.350h 2.371+1 ;

2.30 1.3558 ' 2.Lt158 ,I,. 'l ~

2.32 1.}609 2.b576 2.34 1.}658 2.4996 2.36 1.3705

l 2.5417

2.38 1.3750 2.5840 2.40 1.3792 2.6264

RM-1743 7-2-~6

- 5 -X

3 X3 15 J ..2::__ dx X 'Ji ex-1 X e -1

0

2.h2 1.3832 x10° 2.6689 xlo-1

2.44 1.Jt7l I 2. 7116 I

1 2.,!J6 1. 3907 I 2.7)44 I 2.48 1.3941 I 2.7973 2.50 1.3973 ~ 2.8IJ02

2.52 1.4003 2.8t533 2.54 1.bJ30 2.?265 2.~6 1.4':156 2.9698 2.58 1.4060 I 3.0131 2.60 1.LtlC2 l 3.0565

2.62 1.4122 ).0999 i I

2.61~ 1.4139 3.Et34 I 2.66 1.hl55 ).1870 ! 2.6b l.U.lt9 3 .. 2306 I I 2. 70 1.4181 v 3.2743 j

2. 72 1.hl91 3 • .3180 2. 74 1.1J200 3 .. 3617 2.76 1.4206 3.LC54 2.78 1.4211 3.h492 2.[0 1.4213 ~' 3.4930

2.82 1.4214 3.5368 2. 8L; 1.4214 3.58o5 2.86 1.h211 3.621.!) 2.88 1.h207 3.6661 2.90 1.h201 3. 7118 ~·

2.92 1.~193 3.7555 2.911 1.4184 3. 7"92 2.96 1.1.!173 3.8429 2.98 1.4161 3.8865 ).00 1.L1L7 ~ 3.9301 ~~

3.02 1.1.;131 I 3.9737 3.04 1.hll4 i 4.0172

3.06 1.LC96 1 4.0606 3.08 1.4075 I l-+.101.1.0 3.10 l.~C'54 } 4.1L~ 73

3.12 1.4031 ' !+.1906

j 3.14 1.4006 i 4.2337 1 3.16 1.3980 ' 1+.2768 3.18 1 • .3953 I 4.3196 3.20 1.3924 t 4.362 8

RH-1743 7-2-)b - 9 -

x3 15 X x3 ,

~ r -- a..x:: X X

X e -1 e -1 0

3.22 1.3895 x10° 4.4C.56 x10-1

3.24 1. 3b64

l 4 '\ pl

1 elJL.!.u~

3.26 1.3b31 4.4910 3.2B 1.3797 4-~33'::> 3.30 1.3762 1+.5760

3.32 1.3726 1h6183 3.34 1.J6c9 4.66C.5 3.36 1.3650 I 1.+. 7C26 3.36 1.3611 i h. 7lJL;6 3. 110 1.3570 -!t 4.7b65

3.L~ 2 1.3528 4.8282 3.1.~~ 1 4~-.3 '\._:::> 4.8698 3.Ltb 1.3h41 4 9, 1 " • ...__._c.

3.40 1.3396 ' 4 9r:;r:,.( l

I . ... '-'- } 3.50 1.3350 ~ l+.r:'938

3.52 1o3303 5.03it8 I 3.54 1. 32.55 5.07'::>7 3.~6 1. 3207 5.1165 t

l

' 3.58 1.315'1 i 5.1570 I 3.60 1. 3106 i 5.1975 l

3.6:.:: 1.3055 5.2378

j 3.64 1.3C02 5.2779 3.66 1.2949 5.3179 3.68 1.2895 I 5.3577 ! 3.70 1.2841 l 5.3973 'i"

3. 72 1.27e5 5.LJ368 3. 7L; 1.2729 i 5.4760 3.76 1.2672 5 51 C:? I

. _ _..,.,_

3.78 1.2614 J

5.5541 3.3c 1.255t) 5 -~ 529 !

~

3.b2 1.21;97 i 5.6314 I 3. til~ 1.2h38 I 5.6698

3.86 1.2 378 l 5. 7000 I 3.Bo 1.2317 5. 71+61 } 3.90 1.2255 5.7839

3.92 1.2194 ; 5.8"1t 3.9!.) 1.2131 ! 5.8590 I 3.96 1.2068 i 5.8963 3.;18 1.200.) 5.9334 4.00 1.1941 I 5.9702 'f r

.rr·i-l'7u3 7-'::.-':;6 - 10 -

x3 X 'l

15 5 x-'

dx

" --X -1 }C -l X e t(t e

0

4~02 1.1876 x10 0 6.0069 -" x10 ..~..

4.01..; 1.Hll2 6.0434

j bOb 1.17/it: 6.0797 4.08 1.16tl1 6.1157 4.1C 1.1614 6.:!516

4.12 L15~to 6.1e73 l.1tr 1.1~~cl 6.2227 4.16 1.ll-t14 6.2580 I 1~.18 1.13~7 l 6.2930

~ 4.2Q 1.1279 t 6.3'::_79

4.22 1.1211 6.3625 I

4.2h 1.1143 6.3970 J . L.26 1.10?4 6.h312 l /

4.26 1.1005 l 6.1.;652 ! t

!+.30 1.0936 ~ 6.h990 'f

~.32 1.0867 6.5325 Lt. 3h 1.0793 6.5659 I 4.36 lo,'J72'6 6.59?0 I 1~.38 ""' -.,f.. rc t 6.t:32J -· ,.-' J)0 I

4.l+c 1.0S8t 6.661-~.7 t

4.42 1.0518 6.6972 b ),1,

• "-+-~ l.r:~~~hS 6.r295 4.1J6 1.0378 6.7615 4.48 1.0307 6.7934 ! 4.50 1.023'/ 6,825c l

'f

4.52 1.0166 6. 8561~ h.54 1.())96 t 6.F:E76 4 •. 56 1.0CJ25 6 .. 9186 4.5t. 9. 951t2 x10 -1

6.9 1~94 J~.bC 9.8d3U ~ 6 0790 ~ • / I ,- /

u.62 9.812b 7.0103 4.6~ 9. 7!+19 7 .0 1.;0h 4.b6 9.6711 7.0703 h.b8 9.6Q04 ; 7.1000 it. 70 9.5297 ; 7 i ?01 t 'f o.J..~7t.+

h. 72 9.h590 I 7.1586 /

~~. n.~ 9.3bb4 I 7.1877

I 4.76 9.3178 I 7.2165 4.78 9.2lt73 I 7.2451 u.Bo 9.1770 I 7.2734 'f +

RH-1743 7-2-56 - 11 -

x3 15 X x3

~ dx

X -1 ~ x-l X e e

_, _, 4.82 9.1067 x10 _._ 7.3016 xlO ~

4.84 9 .o 365 I 7 .3~)95

1 1· f:.J-., 8.9b64 I 7.35?3 -+• ........ v

h.to 8.C?61+ I 7.38l.!8 J 4.9C' 8.8266 7.4121

4 qr, 8.7569 I 7 .h391 • /c._ lt 01! 8.6873 7.4668 i ~· / .. -+ I I h.96 8.6179 I 7 .!(926 4.98 8. 5t~87 7.5191 i

I t s.oo 8.1-t 796 i 7.5!.~53

5.02 8.1t107 7.5713 c; nt, "" .... ,-+- 8.3~20 7.5971 5.G6 8.2734 7.6227 s.os 8.2051 7.6t.~B1 I 5.10 8.1370 ~· 7.6732 l

't

5.12 8.ot)n 7.6982

I 5 .1:-i 8.oolw 7.7229 Sa6 7.9340 7. 71-+ 75 I 5.18 7.3668 7. 7718 l 5 -:;n 7.79'77 ~ 7.79':Y9 .. ~ ..... t

5.22 7.7330 7.8195 I

5.2Lt 7.6665 7. 3!, 36 I 5.26 7.6003 7.8671

j 5.25 7.5343 7.8904 5.3tJ 7.4656 't' 7.9135

5.32 7 .h032 7.9364 5.34 7.3380 7.9591 5.36 7.2731 \ 7.9816 5.38 7. 2086 i 8.0039 5.ho 7.1442 I 6.02W ~· ~

5.1t2 7.0802 8.0479 5.44 7.0166 5.0696 i 5.46 6.9532 8.0911

l 5.48 6.5901 8.1124 5.5C· 6.5273 ·~ 8.1336

5.52 6.7648 6.1545 5.,S'L 6.7026 8.1752 ~.56 6.6408 8.1958 5 • .58 6.5793 1 8.2161 5.60 6.5181 6.2363

P!~-l7b3 7-2-56 - 12 -

x3 15 X x3

[ Ji --, dx

X X-.J. X e -1 e

5.62 6.1.+573 x1o-1 t\.2563 xl0-1 5.64 6.3968

j 8.2761

1 5.66 6.3366 8.2957 5.68 6.27h5 8.3151 5.70 6.2173 8.3343

5. 72 6.15e2

I

8.3534 :>.74 6.0994 8.3723 5.76 6.0409 8.3910 5.7S 5.9828 I B.h:.J?? I s.so 5.9250

I

8.4276 I

i' '(

5.82 5.6677 8.'446'J 5 Ql av4 5.8106 8.1-t6.~0 5.86 5.75h0 8.)+618 5. 88 5.6976 8.b994 !

I

5.90 5.6417 8.5169 I

'i

5. 92 5.5861 8.53t~2 ~>.94 5.5309 8.5513 5.96 5.47W s.56e2 5.98 5.1.+215 8.5B5c 6.00 5.3674 \l 8.6016 t

6.02 5.3137 I s.tHn I 6.01-i 5.2603 I 8.6343 6.06 5.2072 I 8.6505 6.08 5.151:6 I 8.6664 6.10 S.1023 '} 8.6822

6.12 5.0504 8.6976 6.11-t 4.9989 s. 7133 6.16 4.9477 8. 7286 6.18 4. f5970

'~ So 7438

6.20 4.8465 6.7588

6.22 4.7965 8.7736 6.2h '4· 7h68 8.7884 6.26 4.6975 I 8.8J29 6.2S 4.64b6 l

8.8173 6. 30 4.6()(',1 J 8.8315

6.32 4.5519 8.8456 6.Y+ 4 • .5041 f.8596 6.36 4.1+567 8.8734 6.30 4.4096

I 8.8870 6.1+0 4.3629 ~' 8.9005

RH-1743 7-2-56 - 13 -

x3 15 X x3

! --dx X ~ X

X e -1 e -1

6.42 4.3166 x1o-1 8.9139 _l

xlO ~ 6.!~4 4.2707

I 8.9271

J 6.l..~6 4.22)1 8.%02 6.J.~o 4.1799 t 8.9531 6.50 4.1350 8.9659

t.52 4.0906 I E. 9766 I I

6.)4 4.0I-t65

l 8.9911 I 6.?6 4 ()0"7 9 .oo :5 ·~ t::. I 6.53 3.9593 9 n1~A

J .J ..... _....•\....

6.60 3.9163 9.0279

6.62 3.8737 9.0399 I I

6.64 3.8314 9.0518 I i l 6.66 3.7895 . 9.0635 1

6.bt: 3. 7b7q l 9.0751 i 6.70 3.7067 ~ 9.0866 '}

6.72 3.665£ 9.098C' 6.71-t. 3.6253 9.1092 6.76 3.5852 9.1203 6.78 3.5454 9.1313 6. 80 3.5060 'f 9. ]_I~ 21

6.82 3.4669 9.1529 6.84 3·'-~282 9.1635 6.86 3.3898 9.1740 6.8b 3.3517 9 .1(44 6.90 3.3140 v 9.1946

6.92 3.2766 9.2C48 6.9Lt 3.2396 9.2Ut8 6.96 3.2030 9.224 7

I 6.98 3.1666 9.2345 ! ! 7.00 3.1306 t 9.2442 t

7.02 3.091.~9 9.2538 7.04 3.0596 9.2633 7.06 3.0246 I 9.2727

l 7.08 2.9899 J 9.2819 7.10 2.9556 9.2911

7.12 2.9216 9.3001 I 7.14 2.8879 9.3091 7.16 2. 851.+') 9.3179 I

I 7.18 2.8214 9.3267 I

! I 7.20 2.7287 ~· 9.3353 f

RH-1743 7-2-56 -14 -

x3 15 X x3 s -dx X ~ X -1 X e -1 e

0

7.22 2.7563 xlo-1 9.3438 _,

xlO-'-7.24 2. 72)~2 9.3523

j 7.26 2.6924 9 .36J6 7.26 2.6689

~ 9.36e9

7.30 2.6297 9.3770

7.32 2.59E9 9.3851 7-34 2.56b3 9. 3930 7.36 2.53b1 9.4009 7.36 2.5021 9.4066 7.40 2.4 ?c5 9 .l..~.lbJ 'f

7.42 2.4491 9.!+239 7.44 2.4200 9 .1~311 7.46 2.39l3 9 .!t35o 7.48 2. 362 e t 9 .1+4:::1 ?.5o 2 .331+6 ~ 9.4534

7.52 2.30b? 9.46J5 7.54 2.2 791 9.4676 7.56 2.25.LS

i 9.474~ 7.58 2. 221_,? \ 9.!t815 7. tfJ 2.1920 t 9 .4l~83

1.t2 2.1715 ! 9 .t~95c 7.64 2.1453 I 9.5c16 J

7.66 3.11?3 I

9.5o82 I

7.68 2.0936 ! 9.5147 !

7.70 2.0602 ~ 9.52ll

7. 72 2.0431 9.5274 7.74 2.0152 9.5337 7.76 1.9:?36 9.5399 7.78 1 .. 9693 9.5460 7.80 1.9452 9.5520

7. 52 1.92JJ~ 9.5580 7.~ 1.0970 9.5632 7.86 1.874:) 9.5696 ?.ee 1.8514 9.5754 7.90 1.8286 9.5E"lo

7.92 1.8060 9.51366 7.94 1.7837 9.5922 7.96 1.7ol6 I 9.5976 7.98 1. 79'c18 I 9.6030 s.oo 1. 7lt1 { 9.to81+

P_!,~-1743

7-2-56 - 15 -

":l 15

X x3 X_.,

.[ --dx X -r. X -1 X e -1 n ..... e

8.02 1.6968 -1 9.6136 x10-1 x10 8.01+ 1.675t I 9.618c I 8.06 1.t5L~7 I 9.6239

J 8.Ci 8 1.b31L1 t 9.6290 p , (' 1.6136 9.6340 ve.J-'-''

8.12 1.5934 9.63B9 B.J1 1.5734 9 .t'43f. 8.16 1.?536 9.6466 8 l q

O_j_.J 1.5337 ' 9.6534 8.20 1.51146 f 9.6581

8.22 1 l 05'"' •4,-1 "-' 9.6627 8.24 1.1-l?bb 9.6673 8.26 1.4581 i 9.t?l8 8.28 1.43Y6

I 9.6763 i

B.JJ 1.1_~213 ~~ 9.6807

8.32 1.!t033 9.68::>0 8.34 1.385!+ 9. t~E93 8.36 1.3676 9.6936 s.1e 1.3503 9.6?77 s.Lc' 1.3331 \!. 9.7017

8.42 1.3ltC· 9. ?OW 8.hh 1.2992 9.nm I

8.1.;6 1.2e2s l 9.7J1JO j ' 8 .L~t~ 1.2t6l I 9.7179 8.50 1.24Y8 ,!

9. 7218 ·~

8.52 1.2337 I

9. 7256

1 b. ~Lt 1.2178 9. 7294 8.56 1.2C·21 9.7.331 8.9: 1.H1t6 9.?Jt;t I 8 {,,-.., 1.1'(12 9. 7hC•4 i o\Jv

E.6~, 1.1561 9. 71~0 p ,cl 1.Jl;l1 9.7!:7';; v. \1.~.

8.66 1.12t3 9. 7~1C 8.b8 1.ll16 ' 9.7544 8.70 1.0972 '

9.7578 •t

8.72 1.oe29 I 9.7612 8.74 1.0687 9.?6L!5 8. 76 1.o~,L8

j 9.7678 8.78 1.0L~l0 9. 7710 s.eo 1.0274 9.77w2

RE-1743 7-2-56 - 16 -

x3 15 X x3 r X 4 -x-dx

X e -1 rr e --1

-1 -· 8.82 1.01.39 xlO 9.7773 xlO .... [i. BL~ l.OOCb ...Jt 9.730/t

1 s.e,t 9.87l6 -2 9.783? xlO s.es 9. 7!~46 J

9.7865 8. 9o 9.bl63 9.?695

8.92 9.h895 9.7924 8.94 9.36L!3 9.75l5b 8.96 9. 24C>6 9.79B2 8.98 9.1184 ) 9.8olo 9.co 8.9977 v 9.6c;s '¥

9.02 8.8754 9.8C66 9.Cho 8. 76;)6 9.8093 9.06 8.t.l.i42 j

9.512C 9.08 8.5293 t' 9.81 1+h 9.10 8.)-!158 9.8172 \1

9.12 8.3036 9.619E 9.11-l. 8.1928 9.b224 9.l6 8.0D34 s;.e21.!9 9.1(3 7.97;,4 I 9.5273 i 9. 2~~: 7.868b ~ 9.82?8

9.22 7.7632 9.8322 q "4 , oL 7 .b?:n 9.5.346 9.26 7.~5b3 9.8369 9.20 7.1.647 I 9.[3~2 ' ; 9.30 7.3544 t 9.UL~l5 '4'

9.32 7.2554 9.01.~37 9.3L'. 7 .l5'(b

! 9. t·1!6C

9.36 7 .CtlO i 9. S!.~[; c ' 9.38 6.9657 I 9.S5C3

i

I ! 9.ho 6.87J-5 t 9.8524 y

9.42 6.7785 9.8545 9.44 6.6867 ; 9.6566 i

9.46 6.5960 i 9.sse7 9.48 6.5065 I 9 .8W7 9.50 6.1~.181 ~ 9.8627 " 9.52 6.3308 9.86L!6 9.51~ 6.2446

i 9.8bbb

9.56 6.1596 9.8685 9.se 6.0756 l 9.8704 9.to 5.9926 9.8722

p..;,:-1743 7-2-56 - 17 -

x3 15 X x3 4 s -- dx

X X -1 X e -1 Tr e 0

9.62 5.9107 -2 9.87140 x10-l x1C> 9.6!~ 5.8299

J

9.f;7)9

1 9.66 5.'(501 9.b77t 9.68 5.6713 9.8794 9. 70 5.5935 9. 8~:11

9. 72 5.5168 I 9.E828 I i 9. n~ 5.L~b-)9 i 9. t'8 1~ S !

5.36c'1 ! 9.8362 I

9.76 I

9.7o 5.2923 I 9.8h78 I 9.8c 5.2194 '{ S'.E095 ~

9.62 5.1474 9.8911 9.54 5.07t3 9.6726 9.El 5.C)Ob2 9.h9!~2 9.2-e h.9370 t 9.6957 9 on 1+eE6E'1 9.8972 \.}; . / '-'

9S2 4. 8013 9.8987 9 .9!.1- 4.7347 9.90C2 9.96 4.6690 9,9016 9.98 4 t0 1 " • . c.f,C i 9.9031

10.CO 4.5402 ~ 9.901.5

x3 15 00 x3 4 s -- dx

y X X -· 1 e -l e -..L TT

X

,.., x1o-3 10.0 h.5402 x10-.:. 9.5500

10.1 ).+.2326 s.e?4E,

1 10.2 J.9JL47 1 t.2h55 10.3 3.6753 7.6590 10.4 3.4234 ?.JJ 27

10.5 3.lc7c 6.6038 10.6 2.96'(b 6.1301 10.7 2.7619 5.6c91 10.8 2.5690 I 5.27cti 10.9 2.3904 t 4.8970

RM-1743 7-2-56 - 1S -

x3 15 00 'YJ

~ Tt _._.- dx

X x-1 X e -1 Tr e

,... xl0-3 11.0 2.2230 x10-c. 4.5~20

11.1 2.0668

j 4.2ll5

l 11.2 1.9212 3.90!.l9 11.) 1.7853 3.6196 11.4 1.6557 3. 35L~6

11.5 1.51~07 3.1054 11.6 1.~307 2.8797 11.7 1.3284 2. 66'(J 11.8 1.2330 ,;, 2,4702

'f 11.9 1.1443 2.2872

12.0 1.0617 t 2.1175 12.1 9.2L~91 xlo-3 1.,;9605 12.2 9.1Y~6 i 1.8139 12 • .3 B.Lt 7C2 t 1.6781.; I

12.4 7.8526 1.~S2 8 ,.

12.~ 7 o2 786 1.4363 12.6 6. '(!_t:) 3 1.3281.; 12.7 6.2h99 1.2284 12.8 5.7896

i 1.13S7 12.9 5.3626 1.0G99

13.0 t.~-.9660 9.7Citl -l· xlO '4

13.1 4.59'19 8.9661 13.2 4.2563 8.2867 13.3 3.9395 t 7.6559 13.4 3.6456 7.0722 "J

13.5 3.3731 I 6.5321 13.6 3.1204 6.0324 13.7 2.61562 I 5.5701 13.8 2,6692 ~ 5.1426 13.9 2 .46ED 4. 7473

14.0 2.2817 4.3817 14.1 2.1091

I 4.0438 14.2 1.9493 3. 7315 i 14.J 1.8013 3.h429 I

~ ' 14.Ll 1.661+4 3.1762 'l

14 ~ 1.5376

l 2.9298

j ....

14.6 1.w2o2 2.7022 14.7 1.3117 2.4919 14.8 1.2112 2.2978 14.9 1.1183 2.1185

PJ-1-17L3 7-2-56 - 19 -

x3 15 00 x3

£ -dx -r-X X Tl~ e -1 X e -1

1s.o 1.0324 x1o-3 1.953J xlo-4 15.1 9.5298 x1c-4 1.Boo2 15.2 8.795l l 1.6592 15.3 8.1165 1.5290 15.4 7 .ut~n 1.4090 f

15.5 6.9093 ! 1.2982 I 15.6 6.3735 I 1.19W 15.7 5.8786 I 1.10]7 i 15.8 S.Ll215 l 1.01!-+7 .,

~ _r 1?.9 4.9993 9. ;L~52 x1G ./

16.0 4.0099 8.6257 16.1 4 "l or' 7.9240 .~4/.::;>

16.2 3.n·r~ l 7.2956 16.3 3.6105 j 6. 7163 16.4 3.3274 f 6.1824

' ~ . - ~

16.5 3.C'662 5.6904 16.6 2.82~·1 5.2371 16.7 2 .(::{J2 7 4.El94 1b.t 2.3976 {· 4.LJ346 16.9 2.2084 4.0501 ~

17.C 2.031~0 3. 753'1 17 .l 1.8731 3.4532 17.2 1. 7247 3.1762

J 17.3 1.S8bO 2.9213 17.4 1.4619 2.6866

17.5 1.34.?7 2.Lt705 ' I

1 I 17.6 1.23c7 2.2716

I 17.7 1.1400 2.o8Et 17.t. 1.0hYl 1.9201 17.9 9.6536 x1o-5 1. 76::>1

18.0 o.E821 1 1.6225 ! 1e.1 8.1756 I

1.l~912 18.2 7 .51'12 1 • .3705 18.3 6.9146 ~ 1.2594 i 18.4 6.3)97 1.1573

.....

16.5 5.8480 1.0633 ~ 15.6 5.371::5 I 9.7695 xlo-6 18.7 L.94.?6 ! 8.9751

1 18.t h.5471 i

13.2446 l 1L.9 4.HD4 f 7.5731

\ \ I

PJ-1-1743 • \ !

7-2-56 \ I

1, 1 - 20 -•., _ _., I

x3 15 f x3

~ -dx

X X X e -1 e -1

X

19.0 3.8429 _c:

x1o / 6.9557 x10 -6

11.1 3.5324

1 6.)881.

l 19.2 3.2468 5.8665 19.3 2.9839 5.3870 19./.j 2.7422 4.9464

19.5 2.5198 I 4.5415 19.6 2.3153 I 4.1694 19.7 2.1272 I 3.8276 19.8 1.9542 3.5136 19.9 1.7951 i 3.2251 I

20.0 1.64tJ9 ~ 2.96)0 J