386 
PROFESSOR G. H. DARWIN ON FIGURES OF 
Performing 2 on both sides, and substituting from (23), 
ft, - 1 = | (2)*T 
«\ 3 Al\ 3r = c0 i = 00 r + i \ r + k \ 7 ;_1 T 
r -1 
+ (f) 2 (-) (-) 2 2 
c r = 2 i — 2 r\i\Tc\r\ i — It — 1 
&/• (25) 
Now let 
r = oo 7, , ,, [ TV — 1 
ft r ) = ,?,Tm'rri 
fti > r] = T r + i!r + iir ’“ 1 1 
t \ i \ k \ t \ r — 
And (25) may be written 
■-o 
(26) 
*• = 1 + * (;) ft r ) + (i ) 2 (?)' ( 7 )' ‘I" ft 0 & A -' 
• (27) 
By imparting to k all integral values from 2 upwards, we get a system of linear 
equations for the determination of the h’s, and it will appear below that as many of 
them may be found numerically as may be desired. 
We now have to consider the series (26). 
Let 
and denote the operations 
B = 
i - r 
i <i k 
A ! cly 
J or 
1 . 
-r* by F; 
\ ! eZT* y 
Consider the function y ] A\y log (1 -f /3). 
Now 
log (1 + P) = - log (1 - y) = 2 7 
Therefore 
Thus 
EK y log (1 +/3) = 
r — 1 
1 cd “ yi-+r + r! 7 r 
A;! cV'= 2 t — 1 = 2 & ! t ! r — 1 
(i,y) = Wylog(l + /3), (i.r) = ^.rlog(l + 5). . . ( 28 ) 
Next consider the function y l E k E l . y log (1 + yS). 
As before, 
r = co iH-_r!_7 r 
E\ y log (1. + /3) 
and 
= 2 *£! r! r — 1 
1 r lk r = oo i r l + r 
FJEK y log (1 + 13) = ~ ~ t 1 
‘ & v ft! cfry* ,. = 2 i! r ! r — 1 
— v 
- 50 i + r! k + r ! 7 ’' 
i ! t ! t ! ft! ?• — 1 
