OF HEAT IN ELLIPSOIDS OP REVOLUTION. 
139 
\ (n—m + 2) (n—m + l)(n+m + 2)(n+m-\-l) (pi — m) (n—m—1) (n + m) (n + m — 1) 
1 0+l)(2% + 3) 3 (2w + 5) ‘ (2» —3)(2?i —l) 3 (2w +1) 
{ 
(2?i—3)0—l) 3 (2w+l) 
(pi —m ) (n — m — 1) (n + m) (n + m — 1) 
} 
(2 n - r o)(2n - 3) (2 n - l)\2n +1) (2n + 3) 
13. The roots of the equation in k being thus approximately found, I proceed to 
show how to calculate the coefficients. When we consider that the series for S-J 
reduces to its first term when r= 0, and to the term a r Pwhen k=n(n-\-l) and we 
suppose the ellipsoid to become a sphere, it is clear that, for the first root, a 0 must be 
the leading coefficient, and that for the (r+ l) th root the coefficient a r must be the 
leading one of the series. Taking the general case first, we observe that, since none 
of the expressions for —, — . . . , — contains it follows that a 0 contains no terms 
«o «o a o % 
lower than a x none lower than U -1 . . . and a r _ l none lower than e, it being under¬ 
stood that a f is finite and of the degree e°. It is also true, as we shall see, although it 
is not so evident at first sight, that a r+l is at least of the degree e, ci r+2 of degree e' 2 , 
and so on. We shall work out the first three coefficients on either side as far as e 3 ; 
though in the subsequent reductions the coefficients of e 3 in a r+l and a r _, are too 
complicated to be worked out fully in general. 
We have seen that 
a. 
Pilh ■ • • PP 
PlPdh ■ • • Pr+ 
— 4 > o • • • <Ml— rS 1 .e 3 + i S 2 .e 4 . . . ). 
But we have also proved that 
and that, up to e 5 , 
bence, putting in this value in the expression for a r+l , 
