254 



Trans. Acad. !Sci. 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, 



(23) 



i — 



.2 



y = 



3 ~3" 



2 i i 



m which u = ~o-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), 



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



3 ~ 3 



as follows, 





w 



+ 



u* 



4(?i + l) 32(?i + l) (n + 2) 



384(n + l)(n+2)(n + 3) 



■} 



On substituting in (24) the values of Jj^(w) and J" i(w) 



"3 ~ 3 



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

 efficient of 6\ which does not contain x, while each term in 

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

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

 putting a; = 



and Eq. (24) reduces to. 



2' = ?T *^"^~^" 



(25) 



