PROFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
511 
These equations give 
¥ = 
and the condition 
]. 1 
y ^ € 
X_ V 
Q — — - - H — — "—’ 
^ / >>/ 6 e 5 e e 
^ ? 
I yf/ e 
af f-V' 
Now 
(1 - 
1—1 
+ 
I 
1 
+ /^)( 
T 
— 
/ 
e 
r 
) = 0. 
76 _ 
I + /3 - A 
/ / 
76 
1 
4 
8 
4 A 
1-8 ’ 
" 
1 
— 
/3 ’ 
e 
- I - 3/3 + A 
e' 
— 
1 
— 
3/3 - 
r “ 
1 + 8 
r “ 
1 
4 8 
Since these values satisfy tlie condition amongst the coefficients, the assumed form 
for ¥’■' is justifiable. 
I find then 
F = 
Whence 
1 + A' A ~ 2/S 
2A ■ 2(1 - /3)’ 
G = - 
I + A A + 2/? 
4A 3(1 - /3)’ 
H z:-- - 
1 + /3 
3 (1 - /3)- 
•> ,2 
O-.C" 
‘ ^ ^ \iMed4') + 1 bl- + 1 ■ (57) 
2A 1 -f ^ 
4A 1+13 
- ?r 
This is the required expression for in surface harmonics. 
Next assume 
“ + ^^1- 
^2~ (^) = <^ + Cl'; 
- A + 1 4 )S 
If we put 
we have 
JcHv- - 1) 
^■2 (</>) == 4 > + Co 
2(A - 1) 
' /3 
€/ = - 2, 
Cl = 
Ci' = 
and 
.8 
A - 1 4 3/3 
3/3 
(g- — l)siir + y)(eisiffi(/) + + (Ti(ay'^ + C) (e/siir </> + {/) + 
Whence F-^, G^, IT^ have the same forms as before, and the condition to be satisfied 
by the coefficients is 
761 _ I li _ fL _ 0 
a'r/ ri ?i' 
It will be found that the condition is satisfied, and that 
1 + A 
F = ^ 
J 6 A ’ 
G. = 
^ 12A ’ 
= - i- 
