446 MR. A. R, FORSYTH ON INVARIANTS, CO VARIANTS, AND QUOTIENT- 

 Thus the quotient-equation becomes 



27s /8 e 2 = ISes'sV = 27s" 3 , 

 and therefore, if be not zero, it is given by 



. - W- 



m \fj. m 



(cz 



which is practically the first integrable case of 83. 



89. The integration of the original differential equation, as given in 86, depends 

 on the supposed knowledge of two special solutions of the equation (24); and the 

 formulae are, for the cubic, the analogues of those quoted in 80 for the quadratic. 

 It is not, however, necessary to suppose two special solutions known in order to 

 obtain the primitive ; this primitive can be derived from a knowledge of a single 

 special solution cr. For we have 



= Mjo-"' + 3u\<r li + a 1 3u\, 



= j (a-" 3o-'0) + 4w ! 1 o- m -f 2o- u 3u\, 

 and therefore 



4o- i r" i - 6<r" 2 



whence we may infer 



V 

 or 



1 = 

 and 



and an expression for u s can be deduced by the ordinary method, for two particular 

 solutions of the cubic are known. 



Similarly, from the single solution T of the quotient-equation we should have 



and an expression for w 2 can be deduced by the ordinary method. 

 90. In connexion with the equation 



regarded as an equation of the second order determining u^ a result, which is rather 

 curious from the analytical point of view, can be obtained. Denoting by p the 



