OF HEAT IN ELLIPSOIDS OF REVOLUTION. 
135 
great; and, in the neighbourhood of these points, the functions . . . «,_ 3 , a,_,, « r _i 
converge with great rapidity. 
The reality of the roots of c o = 0 is thus proved, and the convergency, for them, of 
the series of coefficients. Similar considerations will apply to the roots of 6 y = 0 and 
the corresponding series of 6-coefficients. 
We proceed to approximate to the values of k in series ascending by powers of e. 
12. Although the existence and reality of the roots of a y = 0 are thus established, 
it is a hopeless task to attempt to find them generally. But if we suppose the 
ellipsoid of small eccentricity, and confine ourselves to those values of X which are not 
very great, we may treat X 3 c 3 or e as being a small quantity ; and, if we conceive the 
roots expanded in ascending powers of e, a few terms of the series will be sufficient. 
When c=0 we know from the corresponding solution in the case of the sphere that 
the values of k are given by k=n(n-\- 1), where r— 0, 1, 2, 3 . . . oo ; we are therefore 
to expect that the new roots will consist of series of the form 
&=w(n+l) + ^e+yW 3 +& 3 e 3 + . . ., 
where k x . . . are numerical coefficients. And here it may perhaps be proper to antici¬ 
pate a difficulty which may occur to the reader. The roots, as given above, are 
expressed in powers of X, which are themselves as yet unknown, being determined by 
the conditions to be satisfied at the surface of the ellipsoid. And the values of X 
depend again upon the particular root selected. Thus an apparent indeterminateness 
presents itself, which however is only apparent; and it will be seen, a little further 
on, that the roots form a perfectly regular series, and that we can always choose the 
parts of values of X and k which correspond to each other. For the present therefore 
we shall assume X to be known, and proceed to calculate the various roots of the 
equation in k. 
If we write 0,. instead of k, — k, and calculate successively cq -j- « 0 , a. 2 -t- a 0 . . . 
we find 
el Pi a i a o = 00 
A=0n0i0o 
Pi ■ 1P__ 
0001 </>!</>, 
' Pi 
e"Fu>— <k<k<l>4s\ i— e '( ! I rf / , rf If'r )rf 1 ji* j 1 
^ l \Arfi 0203/ 0,1010203 J 
And, in general, 
Pi 
PlV 3 
*P\P2 ■ • • PrCt r +a 0 = 0O01&5 . . . ^_i(l -e 2 ,_ 1 S 1 +rf,_ 1 S 2 -e 6 ,_ 1 S 3 + . . .) . . (42) 
Wherein 
71 71 71 . 
is the sum of the quantities ~r, TV - • • • T— 
0001 0102 0,-_i0<- 
,S 2 the sum of the products of every two non-adjacent terms of this series, 
,S 3 the sum of the products of every three non-adjacent terms, and so on. 
