OF HEAT IN ELLIPSOIDS OF REVOLUTION. 
143 
(2) r= 1, n = 3, leading term a x put equrd to unity. 
(3) Let r— 2, n— 5 
7 -i n j 7 | 1064 
k = 12 + —-e + „...„ e 
808976 
«o = 
15 1 3f5 3 .7.11 3 7 .5 5 .7.11.13 
e 1 
• • ■ 
18 5.3 5 .13 
o 
792 
2 I 
o e + • • • 
9U0.11.15 
€*+ • • • 
8 
7.5 s ' 3.5 6 .7 
+ • 
^ —30+— ed _—g3 i 
k U + 39 + 3t7.11.13 3 + • ' ‘ 
III. Let m=2. 
(1) If r=0, 77=2, leading term a 0 put equal to unity. 
7 r*. 1 2 o . 20 
* — 6+-e —— e + 0 + • • 
a, = 
oo= 
7 3.7 3 1 3.7U1 
e 5 0 
14 7 3 .11 
e+ . . . 
5 
504 7U5.18 
e s + • • • 
(2) When r= 1, 77=4 
(3) When r=2, 77=6 
7 _ 0 . i 31 1270 2 l 
k - 20 + 77 e ~7UlU3 e+ • * ‘ 
£=42+ 7 — 6-4 
75 . 1534 , 
165 1 11U3.15.17 
e 2 + . . . 
15. The functions with which we have just been dealing belong to the first of the 
two classes of 3-. There is, however, a second class in which 3 is of the form 
3=b 0 P m m+l -b 1 P m m+s + . . . 
The investigation of these functions will proceed on the same lines as those we have 
already treated, and the general formulae for the roots and coefficients will still hold 
true, with the modification that we must here put 77=777 + 2.$+1, where 5=0, 1, 2 ... 
For a given value of m, the first root of each series is 
k= (777+ 1)(tt7+2) + 
3 
6 — 
6(?77+ 1) 
12(m + l)(2m — 1) 
2m + 5 (2777 + 5) 3 (2w + 7) (2m + 5) 5 (2m + 7)(27?i + 9) 
€ 3 + 
(55) 
