1879.] Conduction of Heat in Ellipsoids of Revolution. 9 ( .* 



z being a constant to be determined, the same in both ; and we have to 

 determine the appropriate solutions of these equations. 



Writing v f or cos/3, and denoting by P" t (V) the tesseral harmonic 

 ( n -™>y-/ v 2_ \\j d> ll+>l (r*-iy ^ ig yed that the equation in 6 is 



satisfied by expression of the form 



e^a^Z-a^ + a^:^- ■ • ■ 

 or 1= b -PZ +l - + b«Vl+\ 



and the coefficients (a) and (&) are related as follows — 

 Putting 



\2 C 2 - , (n 2 -m 3 )(>-l 2 -m 3 ) = if w=TO + 2r,and = «,sif n=m + 2s + l 

 (4^ 2 -l)(4.^-12-l) 



2n 2 + 2n— 2m 2 — 1 . / , i \ ^ •£ ,o a ^ -t 



/0 1wo , oS .e + n(n + l)—z=<t)r if n=m + 2r, and =05 if 

 (2w— l)(2^ + d) 



/i=m + 2.s + l, 



we have the systems 



(1) n=m + 2r, /t 1 a 1 =i0 o a o , /t 3 <x 2 =-0 1 a 1 — a- , . . /t r+ i%i = -^^-^-D 



ee e 



z one of the roots of a =0. 



oc 



(2) n = m + 2s + 1 , = -0 O &o> /"A = ~0 A — &0> • ' /V+l&r+i = 1 A — 



6 6 6 



2 one of the roots of b^ = ; 



the expressions dividing themselves into two classes, for the former of 

 which the values of are symmetrically equal, and for the latter equal 

 and opposite, on opposite sides of the equator. 



The values of O also fall into two corresponding classes, and are 

 expressed most appropriately in terms of the function 



sm x 



If we write £=\csinh<z, j^Xccosha, £=cosh<5&, we find for Q these 

 expressions : — 



Class I. Any one of the three forms 



,,,,s4 ? ) + i(.-i)^ (2m+2r+l) ; 2; f +2 ^g- i ) 2m+4) ._ f ) . scg>, 



( - a , v . » 2 r . (m + l)(m + 2) . . . (m + r) a , A 



« (1 p:(?)+l(-i)*«,P(f)- 



1 m 



H 2 



