448 MB. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT- 



92. But the equation (27) is not the only simpler form, consistent with complete 

 generality, to which the original equation can be reduced ; the following transforma- 

 tion presents a close analogy with the reduction of an algebraical binary quartic to its 

 canonical form. From previous investigations we know that, when the substitution 

 z y = u is applied to the original equation, it becomes 



where, if Z denote z"/z', we have 



Z" - 3ZZ' + Z 3 = 



dT 



(these three being equivalent to two independent equations), and 



_ A ^-?i __ Ai p 



fi a O t; - 1 - 9 



. 



25 



The quantity z is at our disposal ; and, if we choose it, not as before in a way to 

 make Ra vanish, but so as to make R 3 vanish, then the differential equation is 



(28) 



which is an alternative canonical form of the general quartic. The equation which 

 determines z is then 



Z" - 3ZZ' + Z 8 = | P 3 - ^ P 3 Z, 



or, writing Z = v'/v, so that vz' = 1, this is 



a linear cubic with its priminvariant = @ 3 . Hence, by the solution of a linear cubic, 

 the general quartic can be reduced to the canonical form (28) ; the new independent 

 variable z is \dx/v, where v is any integral of this cubic equation ; and the coefficients 

 R 4 of the canonical form are then given by the equations 



