H8 DR. J. MERCER: STURM-LIOUVILLE SERIES OF NORMAL 



for a S ft S b, a^t^ A,. Also it will l>e seen, without difficulty, that 



g(s, t, v,n)f(u)du 



converges uniformly to 



\ g(s,t,u}f(u}du 



.'i 



for these values of * and t. 



Retaining the hypothesis that a = *, b = b 3 , it is evident that 



G (*, t, v, n) = \ g (s, t, u, n) f(u) du, 



Jt> 



being equal to 



f g (s, t, u, n) f(u) du- \ g (s, t, u, n) f(u) du, 



.'a Ja 



converges uniformly to 



G (,*,)= [g(*,t,u)f(u)du 



Jv 



for n^s^b, a, s t s />,, ^ v s b. It follows at once that, as n tends to oo, 



f dv\ g(*,t,u,n)f(u)du 



'a Jt 



converges uniformly to 



f dv \ g(s,t,u}f(u)du 



Ja Jr 



tor n s s S b, a, s t s 61. 



II. GENERAL THEOREMS RELATIVE TO THE EXPANSION OP AN ARBITRARY 

 FUNCTION AS A SERIES OF NORMAL FUNCTIONS. FUNDAMENTAL PROPERTIES 

 OF SYSTEMS OF NORMAL FUNCTIONS. 



1. Let *(s,t) be a function of positive type de6ned in the square a == .s- < b, 

 a^t^b. Let i/, (*), /., (s), ..., ^ ( s ), ..., be a complete system of normal functions 

 rf(M), corresponding to singular values X,, X 2 ..... X n) .... It has been shown* that 

 the series 



is absolutely and uniformly convergent, and that its sum function is K (s,t). More 

 generally, the series 



T 

 I'hil. Trans. Roy. Soc.,' A, vol. 209, pp. 439-446. 



+ + .. 



X a -X x-X ' ' V > 



with the 



