Operational Methods in Mathematical Physics. 411 

 Hence, on comparison with (3), we have 



N.-l, *—(£ + £)£. . . (15) 



We can write, as in (8), 



F(p) = 3(c/ cosh </ — sinh q)/q' d 



and then (14) leads to the form 



A(p) = [X(q cosh c/ — sinh q) -f sinh #}/£/, 



where \=K/(c/i). 



The roots a of A(p) = are known to correspond to 

 purely imaginary values of q * ; and they are expressed by 

 the formulae 



q=ieo, ])=—a) 2 a 2 /c 2 = a ) 



where co cot to — 1 — 1/\ = 1 — f/i/K?/ 



and as in (10), only the positive values of co are to be 

 retained. 



It will now he seen that, since A(«) = 0, 



A'(a) = - ~ (X<7 sinh a + cosh (7) = -r— (cos co — Xo) sin w), 



^ y q dp K J J - 2/ 2« v 



while -p, s 3 . 



_£ (a) = - 3 (sin a) — o) cos co j. 



Thus 



F(a) 6a sin co — co cos co 6a 2 sin co — co cos co 



A' (a) co 3 ' cos co — Xcosinco c 2 *co(Xco sin co — cosco)' 



which can be rearranged in various forms, by making use 

 of equation (16). The most compact formula appears to be 



F(a) _6a 2 1 _6a 2 (ch) 2 



Ar(«)~ c 2 ' l-\ + \W~ c 2 'ih(ch-K) + K 2 a> 2 '^ } 



Substituting in (2) from equations (15), (17) we now 

 deduce the mean temperature 



M = & r~(l5 + Mya» + P'-««- c /l( c A-K)+KVr (18) 



The formula (18) gives the extension of (12) ; and agrees, 

 as far as concerns the first and second terms, with A. R. 

 McLeod's result (/. c. p. 143). 



* This is a well-known result and has been proved in various ways 

 in connexion with problems of conduction of heat in spheres. It is 

 deducible from the general discussion given on p. 444 of my paper 

 previously quoted (Pioc. Lond. Math. Soc. vol. xv.); in the notation of 

 that paper, we have to write p=0 in the general formulae there 

 established. 



