254 



Trans. Acad. iScL of St. Louis. 



which is a special case of Riccati's Equation not integrable in 

 finite terms. 



Returning to Eq. (22) and integrating it in series we get, 



X 



2 



^-rr 



(7Jj_(M) + C,J_^{u) 



(23) 



2 i J. 

 in which il = -^x~a^ and J is the well known functional 



symbol of Bessel. Then we have, according to the above 

 theorem, from (21) and (23) as the complete solution of 

 Eq. (19), 



X 



1 ^ 



2 



y = 



rf 



CJ^{u) + 6V_i(«) 



■ X. 



a 



(24) 



Now J ^ and J ^ each represent a series in general form 



3 ~ 3 



as follows, 



"^"^"^ ^ 2T(n + 1) 



W 



+ 



w* 



4(71 + 1) 32(?i4-l) (n + 2) 



W 



384(n+l)(?i+2)(7i + 3) 



On substituting in (24) the values oi Jji^u) and J" j^(?0 



3 ~ 3 



we find in the resulting expression for y a term in the co- 

 efficient of Cj which does not contain x, while each term in 

 the coefficient of C does contain x. Since we know by the 

 conditions of the problem that y = for x = 0, we find on 

 putting x = 



C,= 0, 

 and Eq. (24) reduces to. 



1 





(25) 



