254 



Trans. Acad. ISci. 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) 



JL _ 



,2 



y 



CJ ± (u) + C y J x {u) 



3 ~3 



2 3. 1 



in which u = -^x 2 a 2 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), 



i — 



.2 



y = 



CJ x {u) + C,J_Au) 



(24) 



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



3 — 3" 



as follows, 



J M = 2T(n + 1) 



+ 



4(h + 1) 32(w + l) (n + 2) 



384(n+l)(»+2)(7i + 3) 



On substituting in (24) the values of J x (u) and J 2 (w) 



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 x = 0, 



and Eq. (24) reduces to, 



?/ = -rj^CJ, (U) X. 



J f\ \ K a 



(25) 



