454 MB. A. E. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT- 



quotient-equation, three solutions of the differential equation can be explicitly obtained, 

 and the fourth can be derived. 



Associate Equations of the Qicartic. 



102. The associate equations are two in number. One of these has a dependent 

 variable of the type 



t u lt u. 2 , 3 . /, . . . ...... (33), 



u\, 



u\, 



of which there are four distinct values, so that the equation is of the fourth order. 

 The equation is, in fact, by 52, 



=-zM 



dz 



that is, 



The priminvariants of this are 

 (a) -Q 3 , 



08) Q* - 4 



dz 



dz 



or, since change of sign does not affect the invariantive character, the invariants for 

 the adjoint equation are the same as for the original equation. 



103. The other of the associate equations has a dependent variable of the type 



v = 



u\, 



(35), 



of which there are six linearly distinct values, connected, however, by a permanent 

 bilinear relation 



= 0. 



For the variable given by (35) and the quantities u as satisfying (27), it is easy to 



prove that 



v ui 

 and thence that 



(^ + 4Q 3 v) u - 4Q^ - 2vQ' 4 = - 8Q 3 (u>' 2 - u\u\\ 



