OP HEAT IN ELLIPSOIDS OF REVOLUTION. 
141 
• £L— 
a,_i -f- a, 
a r+2 4-a,= 
€ 
&$r+l.r . 
- 63 ( 
p>- 
iV+1 | 
TV+a , 
D,■+!,,. 
V\ +1 ,r 
H 3 , +1 , A 
Dr +1 .r 
D 
p r e 
p r e 2 8 r _ l<r 
p,.e 3 / 
Pr 
Pr+1 
| Pr -1 
I) 2 ,-!,, 
+ D , r . li ,\ 
D 
H »•+!,»• 
1 d,_ 2i) . 
£0—2 “r~ C£,— 
a,.+2,-7-ci,.= 
1L-) ],D)’+2,i' 
pTpr—\£- 
1 — 
1 — 
S, 
+!,>• 
l, »• . ^>'—2, »• 
« + * 
D,._ \,r D»-— 2 ,«• \ 1,*' 4! r _2 j c 
e 3 
e + • • 
CO-3 -T- £0— 
Dj-.j.j,. D w 
PrPr—\P>-—% 3 
Df—l,r H; — 2 ,«' D ; _g ,. 
.€ 3 + . . . 
S 2 ,+v 
«W 
+ ■ • • 
+ • • . 
• (54) 
The determination of the ratios of the remaining coefficients is thus reduced to 
algebraical manipulation : the last terms, however, of «>._! and ci r+1 are very complicated, 
and I have not succeeded in giving them a simple form, but they may, of course, be 
found in any case where assigned numbers are given for m and n. 
For the others we obtain, on substitution, 
£0-i_i “ £0 
£0+2“ £0 
£0 + g “ £0 
_e_ (2;» + l)(2m — 1) 2 3 , 
2(2« + 3) (2n-l)(2n+S)\2n + 7) e 8 "i" ‘ ‘ 1 
e 2 j 4m 2 — 1 . 3 
8(2n + 3)(2n + 5)~(2^-1)(2n + 3)\2n +3)(2n +Il f ’^ ‘ ' * 
48(2>i + 3)(2a + 5)(2» + 7) + ' ' ' 
Ci ,- ^ . Ct f — 
(n~ —m 2 ) (n — 1 2 — m 2 ) 
(2n + l)(2n-lf(2n-3) 
4m 2 -1 
(n 2 — mr) (n— l 2 —m 2 ) (n— 2 2 — m 2 ) (n —3 2 —m 2 ) 
2 (2a-5)(2a-l) 2 (2?i + 3) 
e 2 — 4. 
8(2w - 7)(2w - 5)~(2n - 3) s (2 n - lf{2n + 1) 
e 2 + <V+ . . 
4m 2 -1 
(2?i —9)(2?i —l) 2 (2a + 3) 
+ • • • 
(n 2 - - m~)(n — l 2 — m 2 ) (n — 2 2 —??i 2 ) (a — 3 2 —m 2 )(w — 4 2 — m 2 ) (w — 5 2 — »i 2 ) „ 
£0_g-£0= 48(2 ?i- 11)(2%-9) 2 (2%-7) 2 (2w-5) 3 (2 ti- 3) 3 (2w-l; 3 (2?i+1) ,e + 
14. We proceed now to the numerical calculation of several of the roots and co¬ 
efficients. 
I. Let m = 0. 
(1) Let r— 0, n= 0 ; « 0 is here the leading coefficient and we may put it =1. 
