74 



C. V. L. Charlier 



Differentiating ouce more the relation (174) we get 



dz^ . 



d^„ 



-r-^ = — cos X — sin x sin x + -r-^ cos x , 



dx 



or according to (173) and (175) 

 (176) 



From (175) and (176) it follows 



ds. . , rf^, 1 

 sm X + cos x=^- 



dx 



dz^ 



sin x 



dx 



X 



(l0^ _ 



cos X 



dx 



X 



or 



f sin , 



^ — dx , 



J X 



and 

 (177) 



y = <]j(x) = cos x(^A — I ~^y~" ^^'^ ) ~^ 



(b + I^. 



+ sm X 



It remains to determine the constants of integration A and B. 

 Developing cos t and sin t into a power-series in t, so is 



cos f df 

 t 



sin t dt 

 t 



log X 



1)" 



2n 1 2n ' 



{2n + 1) 2« + r 



and 

 (178) 



'!^{x) = cos — X -\- 



X' 



X'' 



+ sin X \B log X — 



(179) 



3|_3 



x' 

 2|_2 



For determining A we put a; = 0 and get 



00 



dx 



+ 



5 5 



X* 



TU 



+ ... + 



^ = ^(0) = 



1 + x' 



2' 



For determining B Schlömilch makés use of an analytical method, 

 also perform this determination numerically. Putting x = I we get 



We may 



