150 
PROFESSOR C. RIVER OR THE CORDUCTIOR 
To effect the integrations it will be most convenient to suppose H expanded in terms 
of £ in the form 
B = K*(« 0 P ; ' m m — cqP m m+ ~ + . . . ) (Art. 10). 
But, as is well known, 
w- iy ■ (gpc-- ir-« ■ ■ w 
and may be readily evaluated ; the result we shall denote by j(n, ri). 
We thus find 
1 
j" £l"£l H 'dt ) =%'Z (— Y +s (a r a s -\-a , i a s )j(m +2 r, m + 2s) + %a r .a' r j{m + 2 r, ni +2 r), 1 
r not=s 
JfTVf2”cZ£ may be found by putting n=n', a, = a ,. . . 
For the remaining integral, combine the equations 
fJ 071* 
(1 ■- i") — 2 - Af > n =Wlffl - to 
(1 P.'=X*c ! PP,*-»(«+l)P.*; 
whence 
Yc 2 f £ 2 nP,/d£— (Jc —J— 1 )T n.P w /'c/£+ 
Jn J n 
(76) 
But n(£)=o 0 P."-a 1 P.- +2 +- 
1 
Taking these together, we arrive at the value of j £ 2 o.nc/£. 
Jo 
20. The preceding investigations have related to the laws which govern the move¬ 
ment of heat in an ovoid ellipsoid. For a planetary ellipsoid, we must take 
p=c cosh a sin (3, z—c sink a cos /3 .... 
• • (77) 
The equation which Y satisfies is 
d 2 Y , ePV , J , clY , t n dY , ( 1 1 \d*V <*, 0 n , . ,dV 
~7T+'7R> + tanh a — + cot p—-+ . -~ —= t (cos~ p 4 - smlr a)—, 
do? 1 d{3~ da. 1 ^d/3 \sm 2 /3 cosir ujdfy- t v M 1 ’ dt ’ 
to satisfy which we must put 
V= cos m(f>e 
(78) 
