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



4:71^ — 1 



where /(a;) = 1 — — j-^ — , and y is 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 differential equation (3), 

 and w a solution of the differential equation 



f {x) and ^(x) being arbitrary functions of x. Then 



^,-Vvf{x) = 0, 



Uj ID 



-^^-W4>{X)=:0, 



and, therefore, 



(Pw (Pv _ 



dx^ dx"^ — • w ^) 



But, on the other hand, . 



d'^w d^v d I dw dv\ 



WW a-v a I aw av\ 



V -r-o ^ — 5 = — IV — '^ I • 



dx^ dx^ dx \ dx dxj ' 



hence, 



,.x a / au av\ 



d I du dv\ 



"^ vw 



and 



dw dv r 



(4)' ^■"^~'^^ = J (/-^) ^^^«' + Constant. 



We will proceed to make some applications of the last two 

 formulas. 



If we put 



(5)j v=:VxJJ^ax)^ w = VxJJ^^x)^ 



