532 
PEOFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
Lastly, in accordance with the usual notation for elliptic integrals, write 
Then we have 
A'= 1 — ^—g sill- 6=1 —K ~ sill' 0 
1+^ 
LrzzFr {F.'^y-dd 
( cos - e+^g).A.{i\^yde 
• — iTT 
N=Hf‘' 
J -irr ^ 
( 80 ). 
(81). 
The next step is to express the squares of the P's in a series of powers of cos'- 6. 
When (P''~l)^ it is known that 
P, (/.;u' + (l -pcos (/>) = P, (p) P, (p) +2 'n ^^Pf(p) Pf (p ) cos 4. 
5=1^ r - 
-ji_ g ! 
By putting p = p' we see that '2 . ] (P/(p))- is the coefficient of cos S(j) in the 
expansion of P,( I — (l —p')2 siii'Ti^). Bv Taylor's theorem this last is equal to 
'^'(-)'- «nu.).)d-L no), f.=i 
r=i) 
r 1 
dp*" 
Noav 
d ''' ’’ 
rZp/ 
Also 
ATf (w) (w-1)- terms involving powers of p—l]. 
I ; + r; , 
sin'4</.= 
. V-1 
= 2'(-)'-'.15 
^ = 0 
‘^2'- 
On putting r~t = s, we see that the coefficient of cos S(f) in sin-'’ i<^ is 
(-f 2r: 
' r — ! r 4- 5 P 
