396 
PROFESSOR G. H. DARWIN ON FIGURES OF 
Next put 
and (63) becomes 
i ( A Y 
m = To e \~) pt> 
\ C / 
J& = tW: M 4> 
• (61) 
fa\3 / oO 3 
+ ( 2 ) Ik T} + [ifh ! ‘(jJ‘Y \k,i,r\ y>-^ 
(65) 
We attribute to Jc in (65) all values from co to 2, and thus find a series of equations 
for the [xs . A similar series of equations holds for the M’s. 
We now have to sum the series (62). 
Consider the function 
k + 2 
Hence 
[(1 +/3)* +2 -1] = rr» [(! - y )~™ - 1] = 
k + 2 
1 r = “ k + r\ 
k + 2 r r 2 Jc+ llr— 1 ! ^ 
& + 2 
k + 2 , * + 2 . * + 3 
L 
TT y + 
9? 
/ + • • • 
-1 _ "1°° k + r l 
r = 2 k + 2 ! r — 1 ! 
7' 
7—1 
Next 
1 
{k~ 
{hy} = ^[(l+^-l] 
( 66 ) 
—— . —E_ AJL fy+ir(i + xi x = —±— 
+ 2 ) 7 3 i-V.dy-‘- u L ' TW J» v \i- 1 ! 
1 ^~ 2 r = ® & + r! 
_ V _!_*/+r 
i-2 ^ 
1 r - co 
= 4 2 T 
My-U,r 2 k + 2\r-V. y 
k -\- r\ i r\ 
T 3 r = 2 i — 1! ft + 2! r — 1! r + 2 
!/ + 3 - 
Hence 
1 d l ~ 2 
■ (67) 
M 7 
The differential in (67) must now be evaluated. We have, by Leibnitz’s theorem, 
r ]i — 2 r = £-2 q 0| Vr^.i + 1 /7i 
——Hy +1 r(l+jST +2 — il } = — ——y+14. 2 —-——— 
rffy! -2l7 LWTP; J rfy-G O- r ^ o r! i_ r _2! cfty r cty 
i + 1 ! i + ft — r — 1 ! 7 * 
—(i-r)- M 
•r + l 
i + 1 ! g 5 '=ir 2 l - 2 ! 
3! y r = 0 r! i — r — 2! i — r + L ft + 1! (1 — ^y+k-r 
i + 1 ! 
3! 
7 
7 
r = i — 2 
• (i — v) i+s ,=» 
i 2 . . . fi 
i — r — 2 
Substituting in (67), we have 
l ■ |_ ^ C -f 1 ) 
4AtJ - 6 (Jc + 2 ) 
7 = i — 2 
-1 + (1 + 0Y+* S 3! ,V ~ !t + ;,r 2 irh-n ft-'-" 
' ' ' - r! % — ?’ — 2! ^ — r 4 1! k r 1! 
7 = 0 
.(OS) 
