with a concentric core of a different material 407 



As a matter of fact the equation for A given in the Tripos question 

 arises in the corresponding problem for hnear flow of heat, when 

 the equations for the temperature, with this notation, are as follows : 



du „ d^u ^ , du' ,„ dhi' , 



u = 0, when x = : u' = 0, when x = c. 



u = Uq, when t.^ 0: u' = u^, when t = 0. 



u ^ u' j 



,, du' ' 

 dx ' dx 



, du _ du' I , when x =- b. 



And this problem* can be solved by Fourier's method on the 

 same lines as the solution of the sphere problem which is given in 

 the next sections, a solution suggested by this Tripos question. 



7. With the notation of § 2, the sphere problem reduces, on 

 substituting vr = u, to the solution of the following equations: 



W^^^di^' 0<'r<a...(21), -^ = K^-^,a<r<b... {21'), 



Mi = 0, whenr = (22), i^^ = 0, when /• = 6 ...(22'), 



^1 = "^^o*"' when t = (23), u^ = v^r, when ^ = ... (23'), 



Mj = Wg) when r = a (24), 



1^1 (« ^ - ^h) = K^ [a -^ - u.,j, when r - a ...(25). 



As before 



Uj^ = sin jjLa {b — a) sin ar e~''''^'^\ 



^2 = sin a« sin /xa (6 — f) e""!""''] 



satisfy (21), (21'), (22), (22') and (24), provided that /x^ = ^Jk^. 

 Further (25) is satisfied, if 



F (a) = a sin /xa (6 — a) cos aa + sin aa cos fia (6 — a) 



H ^ sinaa sin ua (b — a) = 0...(27), 



fxaa 



where K^ = K^fxfy. 



This is the equation in a given in § 2 (9). 



Unless )U. (6 — a)/a is rational, the only roots of J (a) = are 

 those of 



a Got aa + cot aa (b — a) -\ = (28). 



'^ ^ ' yuia 



* See also §§ 5, 6 of my paper cited above. 



