320 MR P. ALEXANDER ON THE EXPANSION OF FUNCTIONS. 



If A = 2 (Fourier's Heat, ch. v.; Rayleigh's Sound, ch. xvii.), 



<j)(x) = A (x JgJ-Kfp+tfp M+A-ifa Jfh)~Wp+h( x M + &z-> 



where g , g v g 2 , &c. , are the roots of 



2a JgJ' p+i (a Jg) + (2ah-l)J p+h (a Jg) = , 



which gives an expansion of <f>(x) in terms of spherical functions (Kugel- 

 functionen). 



