148 OR- J- MERCER: STURM-LIOUVILLE SERIES OF NORMAL 



Now we have 

 P f^[/(,-0 + /(- + 0]*-/W = [e->[f( S -t) + f(s + t)-2f(s)]dt-f( S )e->* (35); 



and, in virtue of our hypothesis as to the continuity of /() * it is easily shown that 

 a may be chosen small enough to ensure that 



for all values of/ in (0, a), and for all values of* in (y, 8), the number e being positive 

 and arbitrarily assigned. With this choice of the numerical value of the right-hand 

 member of (35) is less than 



where /is the greatest value of |/(s)| in the interval. Hence, since this is less than 

 e for all values of p which are greater than a certain positive number, we see that 

 the left-hand member of (35) converges uniformly to zero, as X tends to - oo. It 

 follows from what was said above that, as X tends to oo, 



converges uniformly to/(i-) in (y, 8). 



17. Using the notation of 3, we have (vide II., 4) 



-X K x (,, t)f(t) dt = 2 -= ^ (s) (<)/() dt. 



JO n = 



It will be observed that the coefficient of 



on the right-hand side of this equation involves the corresponding singular value, and 

 that, in consequence, its value depends upon the function q and the constants h', H'. 

 The following lemma will enable us to replace the coefficient by another which is 

 independent of both the factors mentioned. 

 Lemma : 



i, X,, ... X,, ... are in increasing order of magnitude, the difference between 



^W^W and "^ (a) *- (t) (36))(37) 



converges unifoi-mly to zero, as X tends to - oo,/or all pairs of values ofs and t lying 

 in the square O^S^TT, O</STT. 



1 To prevent misunderstanding, it may be stated that f(s) is continuous in (y, 8) and in- addition 



