PROFESSOR G. H. DARWIX OX EIJjrSOIDAL IIARMOXIC AXALYSIS. 
473 
differential equations are perhaps the most conveirient, l)nt in the otlier cases a 
reduction seems desirable. 
By considering the forms of and Do in (3) it is easy to show that 
(^'-D 
£ 
Tv 
— 1 {i + t) (w — 1) — D 
- yS 
(m - 1) (w + + 'lv^~ - i{t + _ y 
rr- 
- dct>^ + - /3 
cos — sin -</> + T c<-'S -2(f) +(r 
( 8 ), 
(!>). 
By making ^ vanish we reduce these operators to the forms aj)propriate to 
spheroidal haimonic analysis. By making /3 infinite we obtain the difterential 
equations s})ecified in § 2 as appropriate to ellipsoids of the class c" = + h~). 
It is now necessary to perform the operation xjj,, on typical terms P' and O Ph and 
on typical terms ^^^{sin 
(a.) To Jiad 
Tlie form (8) for is liere conveident. 
It is clear that 
- 1 ) 
dv 
— M / + 1) (w — 1) — .r \ F = (/- — s'-) F, 
because P'' is the solution of the differential e(juation found hy erasing the term 
— .?'P^ from each side. 
Again we have from the same difierential equation 
p -1)1= p' = -2.f; + n^ + i)F+ 
It may be noted in passing that this is equally true when the subject of operation 
is Q^, the function of tlie other form. 
Therefore 
(w - 
dv'^ 
Hence 
V.,(F) (P - .P)F - f3 
-A. 
£ 
— 
i (7 + 
1 ) r" — cr 
V' 
d 
£ + 
a- 1 
T 
+ 
1 + i’-, 
V" 
- 1 ~ 
d 
+ 1) + r 
v~ 1 
- ‘Iv 
Tv 
V- - 1 
P3 
i> 
VOL. (JXOVFI. 
