OF HEAT IN ELLIPSOIDS OF REVOLUTION. 
12 L 
3. For the case of a solid sphere whose radius r— r 0 , these equations become, in 
polar coordinates, 
d 2 Y , 2dV , If, o f /V , 1 cPY] IdY 
dr^r dr + 7’ 2 { ^ dp* dp. + 1 -p* d^ j — f dt ’ 
V=0 or ^ -j-fjV=0, when r=r 0 , 
Y 0 —f{r, /u 
r/E= —dr.r 2 d(j)dfji, where /x= cos d. 
The general type of the solution is 
Y = e~^ il (A cos 7?^+B sin rri(f>) P,/R ;i .(9) 
in which m may have all integral values from 0 up to go, P„ “ cos m<f> is the tesseral 
harmonic of the m th order and n th degree, and If ( satisfies the equation 
d 2 R„ 2 rflt, / n(n + l) 
l^+r *+U- 
If,=o 
( 10 ) 
The two particular solutions of this equation are 
q -J\ IY s in T _,,,A d Y cosXr 
L rfj r ’ A --' UifoJ "V’ 
(as will be presently shown), of which the former only is appropriate to the case of a 
solid sphere, the other becoming infinite at the centre. We have therefore to choose 
R«=S ;i . 
With regard to the integer n, it must be at least as great as m, but may have any 
value from m up to co. The form of the solution being now ascertained, the values of 
X may be found from the condition that at the boundary 
If, = 0 or —— -j-1)If, = 0 when r = r 0 , 
CVi 
and the arbitrary constants A, B . . . may be found from the initial distribution. 
4. We shall now determine the appropriate transformations of (1) and (2) for an 
ellipsoid of revolution, and shall confine ourselves in the first instance to the case of an 
ovoid ellipsoid, reserving that of a planetary ellipsoid for subsequent treatment. 
The axis of z being that of revolution, put x—p cos <f>, jj—p sin <j6 ; equations (I) and 
(2) become 
MDCCCLXX X. R. 
