PROFESSOR O. II. DARWIX OX ELLIPSOIDAL IIARMOXIC AXALYSIS. 
47 0 
Then 
ITphcp 
(•V — 1 V, ^ J _ X J 
i,.{V ) = - i/3 I P' + P' + ^’ _ o^p- + r/, / _ , jp - 
( 12 ) 
(13.) To Jiad yjj, (O P'). 
It Is now best to use xIj^ in the form (5), -where Dp is defined hy (2). 
N o^-i' 
I),(nP') = 
(1 - m 
R - i)(i -13)- 2^] F 
id - 1 
and Dp2 (op ) = (r: _ 1 ) o J [(jR _ ]) (i _ /3) - l>/3] 
r/~F 
IT 
+ ,'i-^ + R-p, 2,/';" - (P4 
v — 1 / dv v~ — 1 \v' — 1 
The latter terms of Ts contribute 
o {-{((+ I) [(R - 1) (1 _ /3) _ [3] F - (.R - ^ct) F} . 
Therefore 
(nP') = n I (p= _ 1) [(p! - 1) (1 - - 2^] i'Y 
+ [(P- 1)(1 -^) + /3]2 
dP' 
dp 
i(i + 1)(1 - /8)(w - ])F 
- RF + /8i (f + 1) P' + j8crP' - 2j8P' ^ 
” 1 
— I 
Ihit 
1) '7'' 
(Ip- 
dp 
/- 
, + + D P' + o 1 P^ reduction that 
dv V' — ' 
V/,(fiP') = !>P'(/5 - i”) - pn 
- <iv R' + i ('■ + I) I" + I (I- + a p- - <tV' 
lip ^ p" — I ' 
()n sidistitutino’ for ■ and ' P^ tlieir values, we have 
^ dp p- — 1 
ii;,(nF) = — I/3n 
O- _ f~\ / 4- a 
P_Lip.' _j_ pi + s _ 2crP'' 
8 
t 
+ [i, — 11P''" 
. . ( 13 ). 
(y.) Wx.({“R^). 
In this case the most convenient form for is that in (9), and we easily find 
