Operational Methods in Mathematical Physics. 413 

 The roots « of A(/?) = for equation (19) are given by 



q = Lco, p= — oo 2 a 2 /c 2 = a) 

 where Jo(»)=0, J 



(22) 



and as before only positive values of co are to be included. 

 We have then 



and8 ° IW-^ 2 W 



A'(a)~ c* ^ ' 



Similarly the roots a of A(_//) = for equation (20) are 

 given by 



q = loo, p = — co 2 a 2 /c 2 = a-\ 



where J (co) -T-Xa>J '(a>) = 1 . . . (24) 



or c7iJ (©) + Kft)J '(ft)) =0J 



and again positive values only are to be used for co *. 

 We have then 



A'(«)=^|{V(?)+Mo(?)} 



and so 



FW = 2 d^ V(y) 



A^j qdqI Q '(q) + \qIo(q) 



_4a 2 J '(©) _4a 2 1 



c 2 J '(co) — Xa)Jo(ft)) " c 2 1 + X. 



^.»2 > 



(25) 



where the final reduction follows from (24). 



Substituting in (2) from (21), (23), and (25), we find the 

 mean temperature 



or 



G i U + 21a;^ + -? 2 i* • w+(K»)>' M 27 > 



where the values of a> are given by {22) or (24) respectively. 

 Of these formulas the former agrees with a result given by 

 A. R. McLeod (I. c. p. 140) ; and (27) agrees, so far as the 

 first and second terms are concerned (I. c. p. 143). 



* That the values of <o are real in (22) and (24) is well-known : and 

 proofs can he obtained from the general discussion given on p. 444 of 

 my paper already quoted on p. 411 above. 



