PROFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
505 
'Fherefore 
eF. - 
since 
•e. 
+ + 1 ) + 
- 4/3(s - 1) + Aie’(o2= - 14S + 12 ), 
- i{i + 1 ) = - j. 
(47). 
Collecting results from (41), (43), (44), (45), 46), and (47), 
{s > 2) €/ = - 1 ) 
+ sV/S'C- + 2t - 1)+ 3{S“ - 2S + 2) + 2T]}, 
7^. (1 - - 1) + 2i6^-[19S” - 130S + 80]K 
= 
ei = 
e. = 
(1 + + 4 ) - - 56j - 180 ](, 
1 + P(i + 1) + 6V./3M5/ + 14y + 12). 
{s > 2) E7= (-)^^^.;i + P(S + 3) 
Ef - 
E' = 
where S = 
+ ^f/3■'[s^SS^ - 2S + 1) - {r- - 2GS - 42) - 2T](, 
f {] + 4/3(S + .3) + + 18«S + 388]}, 
^ + 2 : 
I — 
- '~ZT, {1 + i^(y + 4) + Ti8/32(55/ + 376j + 1044);, 
?:(?:+ 1) „ (i — l)i{i + lll-i + 2) 
6-2 - 1 ’ 
T = 
- 4 
j = i{i + 1) 
(48). 
PART II. 
Application of Ellipsoidal Harmonic Analysis. 
§ 11. The Potential of an harmonic deformation of an Ellipsoid. 
A solid liarmonic, or solution ot Laplace’s equation, is tlie product of two 
P-tunctions of v and of g respectively, and of a cosine or sine function of (^. A 
surface harmonic is a P-function of p, multiplied by a cosine or sine function of (jy. 
We found 
laS") = P’O) + s/3%.,,.P'-'-"(0 + 
where P^ (r) = ) {v~ —l)' ; and a similar formula held for P''(r). 
Hitherto we have supposed P'' (g) to have exactly tlie same form as P'' (v). But 
since g is less tlian unity tin’s introduces an imaginary factor when t is odd, and 
3 T 
VOL. CXCVIl.—A. 
