4 MR. A. R. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT- 



or 



Similarly, we should find 



l -c), 

 = 10 (1 c) 8 (1 c)'. 



A S4 =10(l-c) + 8(l -c). 



105. In the case of the general quartic (for which Q 3 does not vanish) the differen- 

 tial equation for v is 



dz 



= 4 



40,*) 



or, expanded and rearranged, it is 



When the covariants 3 and 6 4 are introduced, this is 



106. A first inference from this equation (36) is that 



v* 



= B 



8> 



where B is a determinate constant depending on the selection of the original funda 

 mental system of u integrals. 

 107. Next, the substitution 



changes (36) into 



V = 



(37) 



dz 



= . . (38), 



where, after long and laborious analysis depending largely upon continued application 

 of the theorems of Section III. as to the values of the successive derived invariants, 

 it may be proved that 



