314 MR P. ALEXANDER ON THE EXPANSION OF FUNCTIONS. 



ever of x and a consistent with convergency, and A («), A^a), A 2 («), &c, and 

 S (x), S^x), S 2 (x), &c., are connected by the following relations :— 



A 1 («)=-J^[A («)],1 



d\ ' 



A w+1 + 2^-A ?l _ 1 = ) 



and 



i 



i 



1 dx I 



dS 



and hence 

 and 

 where . 



A n = (— i) n cosnAi- A , 

 S w = ( — i) n cos n A . S , 



and \ and A are operations denned by 



a d 



% COS A, = -r , 



and 



d 

 i cos A = 7 — • 

 dx 



Konig's method seems to be much more general than Sonine's, as Konig's 

 functions G , G l5 G 2 , &c, may be connected by any law, while Sonine's functions 

 A , A x , A 2 , &c, are connected by one law only. But on the other hand, Konig's 

 functions are limited by the condition that G p (x) must, when x nearly equals c, 

 be capable of expansion in ascending integral powers of (x — c) beginning with 

 (x— c) p , whereas Sonine's functions are subject to no such condition. 



Both methods give the expansion of (p(x) in terms of J (#), Ji(ai), J 2 ( x ), &c., 

 Bessel's functions. But neither of them give the expansion 



<p(x) = A J n (k x) + Ai J„(/ci») + A 2 J n (k 2 x) + &c, 



where k , k^, k 2 , &c, are the roots of some equation of condition. 



The most general method of expansion I have seen is that of expansion in 

 normal co-ordinates employed by Rayleigh throughout his Theory of Sound, 

 which is so satisfactory that had I become acquainted with it somewhat earlier, 

 I would probably not have sought after the following method : — 



The general problem is to determine A , A 1? A 2 , &c, so that when possible 

 <p(x) = A G (x)+A 1 G 1 (x) + A 2 G 2 {x)+&c, . . . . (1) 

 where G , G u G 2 , &c, are connected by some given law. 



