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XV. — Expansion of Functions in terms of Linear, Cylindric, Spherical, and Allied 

 Functions. By P. Alexander, M.A. Communicated by Dr T. Mum. 



(Read 20th December 1886.) 



The expansion of <f>(x) in terms of G Q (x), G^x), G 2 (x), &c, connected by a 

 given law. being of great importance in mathematico-physical investigation, 

 every method of effecting this expansion must have some interest for scientists. 



I therefore proceed to propose what I think to be a new method, in the 

 hope that it may prove to be useful. 



Many special expansions of this nature have been effected by Fourier, 

 Legendre, and others. 



After I had developed my method, my attention was called to two papers 

 on this subject showing methods of development of great generality. The 

 titles of the papers are — Konig, J., " Ueber die Darstellung von Functionen 

 u. s. w.," Mathematische Annalen, v. pp. 310-340, 1871 ; and Sonine, N., 

 "Recherches sur les fonctions cylindriques," &c, Mathematische Annalen, xvi. 

 pp. 1-80, 1879. Konig, assuming that 



<p(x +x) = ¥ (x ) . G (x) + F^) . G,(x) + F 2 (x ) . G 2 (x) + &c. 



where G , G v G 2 , &c, are an infinite series of functions of x, connected by some 

 given law, and also subject to the condition that when x is nearly equal to c, 

 each of them is capable of expansion in ascending integral powers of (x — c), 

 beginning in the case of G p (x) with (x— c) p , proceeds to show that the coefficients 

 F (x ), F 1 (x ), &c, are to be deduced from the following — 



G (c).F (* ) =<j>(x +c), 

 G (c) . F '(b ) = F (x ) . G '(c) + F^G^c) , 

 G (c) . F "(* ) = F (* ) . G "(c) + F^ ) . Gt^e) + F 2 (* ) . G t "(«) , 

 &c, &c. 



Sonine shows that 



8 (a+x) = A (a) . 8 (x)-2{A 1 (a)S 1 (x)- A 2 (a)S 2 (^) + &c.} 

 if the series is convergent, where S (aj) and A (a) may be any functions what- 



VOL. XXXIII. PART IL 3 A 



