OF HEAT IN ELLIPSOIDS OF REVOLUTION. 
131 
It was the consideration of these formulae, including as they do both the cases of 
n—m even and odd, which suggested the transformation (32). 
10. We shall now sum up the results of the last two articles and shall replace, in 
doing so, the symbols £, Q, H by S. Neglecting unnecessary constants, we obtain 
(1) n—m even, distribution symmetrical on opposite sides of the equator, 
£=a 0 P—oqP„ “ + 3 +a 2 P—a 3 P m m+6 + . . . 
P(^) — a 0 S OT 2m + 3 
1.3 
(2m + 5) (2m + 7) 2 ra+4 
1.3.5 
(2m+7) (2m + 9) (2m + 11) 3 OT+6 
I- • (36) 
n (v) 
_( 77 2 —X 2 c 2 ) 2 J q , 2(m + l) 2 2 (m+l)(m + 2) 
o^ -1 " 2m+ 3 1 ^ +2 ”t( 2m +5)(2m+7) 
2 3 (m + l)(m + 2)(m + 3) 
(2m+7)(2??i + 9)(2m+ll) 
%S ;a+ G + . . . 
(2) n—m odd, distribution equal and opposite on opposite sides of the equator, 
^=6 0 P/' i+1 -6 1 P,/ +3 +6 2 P^ +5 -6oP / / i+7 + ... 
m= £± r fh \ 6 0 s. +1 
' 2m + 5 & ‘ S »+s+(2 m +7)(2m+ 9) & » S "+= 
___ h a i 
(2m + 9)(2m + ll)(2m + 13) 3 , " +7 ‘ 1 “ 
2(m + l) , q 2 2 (m+l)(m + 2) 
2m+ 5 1 ' H+3_h (2?/i + 7)(2m + 9) - * +s 
2 3 (m + l)(m + 2)(m + 3) , 0 , . 
+ (2m + 9) (2m + 11) (2m +13) -»+? + 
b. (37) 
It will be observed that all these expressions for fl, notwithstanding the factors in 
the denominators, are necessarily finite at the centre. 
The first equation of last article (19') suggests yet another form for fl which we shall 
find of service in calculating the coefficients. 
77 
Putting £= cosh a = —, that equation may be written 
j^W^a-ksi; 
and a comparison of this equation with the corresponding one (18) in 3- shows that we 
may satisfy it by 
s 2 
