176 DR J. MERCER: STURM-LIOUV1LLE SERIES OF NORMAL 



Employing .(.) with the signification of the preceding paragraph, and denoting 

 by .?. (*) the sum of the first n terms of the series 



I sin i* f/(l) sin & dt+ Z - sin J /() sin |f A 



IT ' 



-" . (29) 



7T 



we deduce at once that, as n tends to oo, 



O-H (s) - . () 



converges uniformly to zero on the whole of (0, TT). The coroUaries to this result are 

 of the same nature as those stated in 10, the only difference is that ,*() replaces 

 ?.(), and (29) the series (25). 



12. We shall state the corresponding results when the pair of boundary conditions 

 is -B', or ;B', more briefly. In the case of the pair *B' it will be found that (27) must 

 be replaced by 



Replacing the boundary conditions (28) by 



$? = at s = 0, u = at = ir, 

 as 



it may be established that 



T x (, ) = 2 7 - j^r r - cos (n-J) * cos (n-) i ; 



= 1 ^71 j) A 7T 



whence, in virtue of the results quoted in III., 19, it may be shown that (30) 

 converges to zero, for unequal values of s and t* Finally, if '?.(*) is -employed to 

 denote the sum of the first n terms of the series 



- cos s [*/(*) cos fy dt + - cos |s f f(t) cos ft rf 



+ ...+ - cos (n-i) s ("/() cos (n-i)i^-f ..... (31) 



7T Jo 



we obtain the result that, as n tends to oo, 



0-, () - * (*) 

 converges uniformly to zero in the interval (0, TT). 



* This may be deduced from the corresponding result obtained in the preceding paragraph. For, if 

 f . (*) is the normal function which, for /i = A,,, satisfies 







and the pair of boundary conditions *B', then, for the same value of /*, ^ n ( - s) satisfies this equation and 

 i pair of boundary conditions of the same typo as oB'. 



