FUNCTIONS IN THK THEORY OF INTKOltAL EQUATIONS 157 



and that their order of arrangement is that in which r. increases with n. With 

 this understanding we shall call (45) a Sturm- Liouville series corresponding to F (x). 

 Applying the transformation of 2, we see that 



when expressed in terms of s, becomes a function, say M*)> which sjitisfies the 

 equation 



00 



and the pair of boundary conditions B', B', "B', ;B' which corresponds to the pair 

 satisfied by , (x) : the appropriate value of p. is evidently r n ^'\ Since (46) leads to 



f [*()] <fe=l (" =1,2,...). 



Jo 



it is thus evident that ^ (*), ^, (s), .... ^. (), .-is the complete system of normal 

 functions satisfying (2) and these boundary conditions. 

 The series (45) becomes 



.1 ot 



where /(*) is F(a;) expressed as a function of *, and w(s) has the meaning attached 



to it in 2. 



23. Let T (x) and A (a^ be the upper and lower limits of indeterminacy of the 

 series (45) at the point x. These numbers are obviously the upper and lower limits 

 of indeterminacy of the series (47) at the corresponding point s of (0, IT). From the 

 results of II, 4, we therefore have 



where K* (s, t) has the signification of 4. Let us suppose for the moment that x is 

 a point of the open interval (a, l>). After what was said in 10, 11, 19, it will be 

 clear that 



Further, 



where d is the upper limit of | w' (s) \ in (s, s + ct). 



