144 
PROFESSOR C. RIVER OR THE CORDUCTIOR 
As the mode of formation of the coefficients has been already sufficiently illustrated, 
I shall confine myself to writing down a few of the smaller roots of this class. 
3 6 4 
When m=0, s=0, 2+ - e — e~ — yyr^e 3 + . . . 
5 87a 3.5°.7 
23 23114 , 
„ m =0,s=l,h=12+- ‘+£gr*n 
. 7 59 696718 0 , 
” »«=o, «= 2, fcbo+jjj r + ■ ■ ■ 
” m=1 ’ *=°> *= ®+ ? £_ iob £ " + nhi £3+ ■ • • 
_ 37 21192 , 
,, m —1, S — 1, k —20 + — e ~h .y ^ 3 11343 e ~~h • • • 
07 , 27 , 73892 
55 m — 1 > S — 2 ’ 42 + 5 h e +3.5 3 .ll 3 .13.17 € ~ + ‘ ' ‘ 
” m—2, s-0, k— 12+ 3 e 891 ^"*"3'.11.13 
-o <* + 
17 
„ m = 2, s=l, &=30 + - e 
506 
39 1 3411.13 
Y s e 2 +.- 
The larger m and n become the more nearly will the first few terms of the series 
represent the value of the root. 
16. We shall now show how to find the types of heat-movement which take place 
when the surface of the ellipsoid is maintained at a constant temperature zero. The 
general equation of conduction is satisfied by any expression of the form 
V=(A cos (3 sin w<£)e -A= bA/(/3)fi,/(a); 
and, in order that the temperature may be constantly zero at the surface, we must 
have 
or, more fully, 
fV(a 0 , X, m)=0 
(56) 
—cqS^+g-f ... —0, 
in which the leading term is (— l)'a,.S w+ o,., where m-\-2r=n. 
For given values of m and n this equation has an infinite number of roots; the total 
number of values of X is therefore, apparently, triply infinite. Now, in the corre¬ 
sponding problem for the sphere, the equation is S„(Xr o ) = 0, the number being doubly 
infinite; and there is no theoretical difficulty in using this solution to approximate to 
