540 
riiOFEShlOU G. II. DAUWIX OX ELLIPSOIDAL IIAPMOXIC AXALYSIS. 
Ill the preceding formula the sum of this last function had limits i— \ t<) 1, hut as 
Ave ]iow see that it vanishes when r=i, the up2)e]' limit may he changed to i. 
It follows that the terms of the second order are 
+ 1 ) — 1 ). 
i-r 
{-y 
i±4n_Li/5r;4(llIl±4'' 
L — J'’. 
Q-2 jx ^ .A PAY ; 
(r + lh ' (r!/0- + l)J 
~ -1/^4 /^{kj ~ i“2)+ 
'fhe te]'3n in in this expression will he found to he — That in Uq will he 
found to he Vac Tlien since ^■2,— —^], «o=l; these terms are together }/ 3 '( 5 /+lt)' 
The Avhole may then he written 
%j 
Now 
jy.!;.+ !! 1!2^ 2! 2.3 ■“ Ij- I— I 
4 , (-)'• ^+H_ _t(i + 1 ) (t+l)(i + 2 ) dt- 1 ) 
T(r + l)V-/-!” 2 ! 2 !‘^ 3! 3! 
+ 1 K — t — 1 ) ( — t) )(t + 2 ) ( — i — 1 ) ( — i) ( — 3 + 1 ) 
2! 1.2 3! 1.2.3 
'(t, —1, I, l)~I + i(i-}-l 
I 
1 ) 
1 
V+ 1 ) 
— — y ^’(h ~ 1 > T I ) + ^. — I 
] 
The last result follows from the fact that in accordance A\ ith (33) the sum of the 
hypergeometric series has an infinite factor in the denominator, and vanish os. 
i O 
Then since hy (32) S a 2 ,/(^’+ f ) = A . > fTe terms of the second order are found to he 
o'" + i 
J lence, collecting terms, 
X 
jj — +^+ 8 /^' {4 + 11)- 
•Suljstituting for 11 its value (73), A\'e have N or 
f' y; ((C. py in = .V [1 - 1 /S+y/ 3 - (/■- 4 y'+ • 2 )]+ 2 :.JI [/ 3 + ' / 3 ' - 1)]. 
• —^77 _ t "t" f 
