S^-lJ-j-f^Ste-BaJ-j-H-*. 



136 Mr. J. Cockle on Transcendental and Algebraic Solution, 



3(fl 8 -l)^=#*+aa?-2 (a) 



Differentiating this, and transposing, 



^ = (2*-5a)| 



Multiplying this result into 3(a 2 — 1), and reducing by means 

 of (a), we find 



3 2 (a 2 -l) 2 ^=(2 t r-5«)(^ 2 + ^-2)+3(a 2 ~l)^ 



= -3<M? a -(2a*+l)ff + 6a; ...(b) 



whence, putting ^=w/—l, 



d-rf)S-5-^-a .... (2) 



the differential resolvent to which, in the Number for November 

 1860, we were led by an entirely different process*. But (2) is 

 equivalent to the symbolical equation 



( ^^ • s ±m ) ( ^p? • £ +™)*=°> 



whence f 



Hence, introducing arbitrary constants and substituting for in, 

 the general expression for x is % 



sin - lo sin -1 a , . _. v 



^=K ie ^ 7 ^ + K 2 e~^^ = Asinr s ^-^ + BV . (3) 



* The process given in the May Number leads to 



dx 2 x x(a — x 2 ){a — x 3 ) a?x-\-ax 2 —2a 



da ""3(1— x 2 ) 3(a — x) ~~ 3(a — x)(a—x 2 ){a — x 3 )~~ 3(a 3 —a) 



which reduces itself to the result (a) of the text. 



f I have found it often convenient to represent by (a] [6) the product of 

 a and b treated as ordinary algebraical quantities. Thus we see at once 

 that 



(vw.^±»][vw.^+»)=(i-«^ 2 -» 2 . 



and the accuracy of the symbolical decompositions given in the text are 

 manifest. 



X If the first coefficient of a linear differential equation of the second 



