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



As each of the three terms within the bracket has been shown to converge uniformly 

 to ro in (y, 8), it follows that, a, n tends to oo, the difference 



s. (*)-?.(*) 



converges uniformly to sen) for values of . in (y, 8). It may be shown in the same 

 way that each of the differences 



has the same property. 



Since (y 8) is any interval lying within (0, IT), it follows that, as n tends to oo, the 

 difference between any two of the sums s. (), ,5, (*), '5. (s), Js. (*), converges to zero at 

 each point of the open interval (0, IT). It remains to consider the end pouits of the 

 interval. At s - we have 



and, of course, ,, () = ;,. (0) = 0. 



From the first two formulae we obtain 



= I i' f(t) cos n dt+ - [*/(*) tan J< sin rf c/t. 



As both f(t) and /() tan ^< possess Lebesgue integrals in (0, IT), the integrals on 

 the right converge to zero, as n tends to oo.* We thus see that 



fim[-si,(0)-s.(0)]=0. 



!> 



In a similar way it may be shown that 



u(w)-9(7r)] = 0, 



!!** 



whilst 



".()=* 5* ()=.0, 

 for all values of n. 



After what was said in the corresponding case dealt with in 10, the reader will 

 perceive the l>earing of the results of this paragraph upon the convergence of the four 

 trigonometric series (25), (29), (31), and (33). 



16. The reader is now asked to review the results which have been obtained 

 above. It was shown in 10-12 that the limits of indeterminacy of any canonical 



*!!., 6. 



