﻿60 



INDIANA UNIVEESITY 



Let be the abscissa corresponding to a point on the curve of 

 the function, and m an arbitrary Avhole number; then 



0 < — a) ^ 1, determines an integer a. 



Let a^xo — a = Xi. Then o <i Xi < i. 



1 ,5 

 Let x' = ; and x" = 



1 . 3 

 -4-^1 2~^^ 

 Then x' — Xo = — ; and x'' — Xo — 



— ij?o 0 {x — Xo) 



^ yn + n (sina''' + ''x7t~sina'^ + "xo7t)2 



« X — Xo 



= A + B, letting A stand for the first summation and B for the 

 second. By choosing m sufficiently 1 arge x' — Xq can be made to 

 differ as little as we please from zero. It is evident that the limit 

 of the above fraction 



f ix') -f(xo) 



X X 0 



as x' — Xq approaches zero is a value of the derivative at x^. 



Our problem is therefore to evaluate this fraction, when 



x' — Xo—O. 



To evaluate A: 



• . / • . 2 cos a'^^^^n- sin a^'-^n 



sm a^x n — sin a^'XoTt 2 2 



a'^ix'-x^) a'^\x'-Xo) 



jt cos- a^^-^^7t. sina^ ^- — 



This expression is less than n in absolute value, since 



— 1 < ^ < 1 and -1 < cos y < 1. 



y 



Substituting this value in the series A we have, 



m — 1 



0 



r-, sin a^^x' 71- sin a'^^Xo 71^ 

 a^{x'-Xo) 



0 ab~l 



