130 
PROFESSOR C. RIVER OR THE CORDUCTIOR 
9. But we may also express 12 as a function of 77 , the major semi-axis of the confocal 
ellipsoid. And, if we turn to the original equation in 11 (19,) and put therein 
rj = \c cosh a, we find 
(rr-W) 
d~n , cm 
vrc 2 \~ 
7 f-XW 
n =—ri z n+kn . 
(19') 
We shall first replace fl by W, where 11 =( 77 s —\ 3 c 2 ) 2 W, the equation becoming after 
substitution and reduction, 
( 77 - —X 2 c 2 )^+ 2 (m+ 1)77 ^+m(»i+ 1 )W= - 77 3 W+ifeW. 
If we now replace W by w where W=r]~ m w, we find after reduction 
dho 
dw 
dr) 
r p” 2 + 2 V or +y*w-hw= Ac 2 ( Xi— T 
0 Jd?w 2m dw m(m + l) 
dr f r) clr) 
w 
and we can satisfy this equation by a series of S-functions. 
Before doing so, we calculate the value of the expression 
d~ S» 2m dS n m(m + l) 0 0 2 .( 777 , +1) dS„ ?i(?i + l)+m(m + l) 0 
7 o 7 \ o ^~7 \ 0 
ar)“ 7} clr] 7 ]~ 7] arj rj-* 
(n + m+ 1 )(n + m + 2) 
(27i +1) (2 ?t, + 3) 
S, H 
2rf + 2*-2m»_l gj+ O r »)q r «-l) g b (29) . 
(2/i-l) (2n + 3) 
(2»-l)(2» + l) 
Putting 
S„= 
1.3.5 . . . (2w-l) 
1.2.3 . . . (n+m) 
H„ 
(34) 
whether n—m be even or odd, we obtain 
d 3 2m d m(m + l) 
dry 77 dr) r f 
H*=H„ +2 - 
2 ?i 2 + 277 — 2m 2 — 1 .. (n~ — m~)(n — l 2 —m 2 ) 
(2n—l)(2n+o) (77-] }(4.»-l 2 -1) 
H„_, 
The equation in ir may, therefore, be satisfied by either of the forms 
and 
w-—a^S- m +oqH w+ 3 +%H W2+4 -|- • • 
W - 5qH w+ j -f-/qH m + g +6 oH, w+5 4- . 
VI 
( V 2 -\W 
n=— - L iv 
(35) 
