OF HEAT IN ELLIPSOIDS OF REVOLUTION. 
145 
the roots of O,/= 0 , where the eccentricity of the ellipsoid is small. The precise 
manner hi which each of the roots of the first problem resolves itself into several in 
the second is interesting. We observe that S,„ +2 ,. may be the leading term of H, either 
as the first term of 0,° m+Zr , the second of the third of 0 2 M+2; ._ 4 , ... We infer 
therefore that each root of S 0 =0 or of S^O corresponds to one for the ellipsoid, each 
root of S 2 =0 or S 3 =0 to two, each of 8^=0 or S 5 =0 to three; and, in general, that 
each root of 
S 2 „= 0 , or ol & 2 «+i= 0, 
corresponds to (n-\- 1 ) roots of the equation in X for the ellipsoid. 
If, therefore, the total number of values of X in the case of the sphere be wN, the 
total number for the ellipsoid is d -~ ^ ^ N. 
All the roots of S „=0 may be found without difficulty by the general processes 
given by Professor Stokes (Camb. Phil. Trans., vol. ix.) and by Lord Payleigh 
(Proc. Math. Soc., vol. v., p. 119); and, therefore, those of ffi/ can be expanded in 
powers of c 2 . AVe shall confine ourselves to determining those of n o "= 0 . 
Employing the expression for O( 77 ) and putting cc 0 = 1 , we have 
^0+t a i^2+V5+ . . . —0 
(57) 
If we neglect e this equation reduces to S 0 (Xa) = 0 , whose roots are given by Xa—dr. 
Let the full value of Xa be ITT ~\~1 “h • • • 5 e being the eccentricity of the 
c 
a" 
ellipsoid = . The elements of the subsequent calculation are briefly as follows :— 
e=X 2 c 2 = f 2 7r 2 e 2 + 2drfi(d-f- . . . 
fie 3 
k d 3 
S 0 (Xa) = 1 cos Itt ~fi ~ —vrv ) cos dr.e 1 -f- . . . 
> yjf \ OTT 0~Tr~ I 
11 r ^~7^' 
S 2 (Xa)= — 7)7 o cos nr — | t-tt—;) fie 2 cos dr + . . . 
ITT V TT 
S*(Xa)=(^-^)cosw+ ... 
2 
567 
8 l 4 TT 4 
.77 a -l~ coc e + . . . 
.3o o2o 
Substituting in the above equation, and putting ecpial to zero the coefficients of e~ 
and e 4- , we obtain finally 
ITT 
Xa=i7r+-dre 2 + 7T7:(i 2 7r 2 +27)e 4 +.(58) 
o 4Uo ' 
MDCCCLXXX. U 
