138 Mr. J. Cockle on Transcendental and Algebraic Solution. 



-JL l 



~ + 2m' (a±* / a?-l) m ' 



Hence, introducing arbitrary constants, effecting an obvious 

 reduction, and substituting for m, the general expression for a? is 



x = Ci t/aTVa^l +0,,^=— 



=Cj v^tvV-l + C 2 v^flq: Vff*=l. 

 By substitution in (1) we are led to 



0=(C 1 3 + C 2 3 + 2)a + 



3(C 1 2 C 2 -C 1 )^a 2 ±^^" 1+ 3(C 1 C 2 2 --C 2 )v / «+V'^l, 

 the dexter of which will vanish, independently of a, if 



C 1 3 + C. 2 3 + 2 = 0, and C 1 C 2 =1. 

 Hence the values of the arbitrary constants may be written 



C, = (-l)*> C 2 =(-l)* 

 and we may put 



These agree with known results. Further, assuming for the 

 solution of (2), or rather of (4), the series 



2a r a r , 

 r being taken from zero to infinity, the form of those differential 

 equations shows that the above breaks up into two independent 

 series, and that we may assume 



D and E being arbitrary constants, and a and /3 being deter- 

 mined by the conditions 



«2r +a (6r-])(6r + l) 

 *2r (6r + 3)(6r + 6)' 



Ar + 3_ (6r + 2)(6r + 4) 

 /3 2r+ r (6r + 6)(Gr + 9)> 



in the sinister of which u and u { may, in consequence of the 

 arbitrary nature of the multipliers D and E, each be taken as 

 unity. Now, when = 0, then 



# = D = 0, or a/3, or -*/3; 



