100 • Trans. Acad. Sci. of St. Louis. 



where / (ic) = 1 — — ^71 — ' ^^^^ if ^^ given by the formulas 



(2) or (2)' according as n is, or is not, an integral number. 



Now, let V be a solution of the diiferential equation (3), 

 and w a solution of the diiferential equation 



(•^^ ^^, i-y/c^(x)=0, 



/(x) and ^{x) beinoj arbitrary functions of x. Then 



dhfj 



-^ + w(f)(x) = 0, 



and, therefore, 



dhv (Pv 



But, on the other hand, 



<Pw (Pv d I dw dv\ 



a I 

 Ix \ 

 hence, 



y ^ W = — ly W 



dx^ dx"^ dx \ dx dxi 



('^ .fa("5S-"s) = (/-*)™' 



and 



(4)' ''~dx~^'lx^} ^f— ^> ^'^^^ + Constant. 



We will proceed to make some applications of the last two 

 formulas. 



If we put 



( 5)j y = V xJJ^ax)., w = V xJ,^(^x) , 



