410 MR. A. B. FORSTTH ON INVARIANTS, CO VARIANTS, AND QUOTIENT- 

 There are two primin variants, viz. : 



3 = 



and there are three proper quadrinvariants, viz. : 



e-u = 6Q 3 Q" 3 - 70?, 



*i = 8 (Q, - 2Q' 3 ) (Q" 4 - 2Q'" 3 ) - 9 (Q' 4 -2Q" 3 )', 



63.4,1 = 



= 4Q 4 Q' 3 - 3Q 3 Q' 4 + 6Q 3 Q' 3 - 8Q?.. 



And if we choose we can replace any one of these by a linear combination which 

 includes that one ; thus we could replace @ 3li ,i by s,*,! BM, the value of which is 



dz 3 dz 



Independently of the special application to the deduction of quadrinvariants, the 

 preceding analysis shows that, when a number of invariants are given, there are two 

 methods of forming new invariants, viz., the quadriderivative process and the Jacobian 

 process. 



Cubinvariants. 



38. We now proceed to apply these methods to obtain the proper invariants of the 

 third degree. The quadriderivative process will not produce any invariants of this 

 degree when applied to any of the invariants already obtained ; and, therefore, all 

 that remains for us to do, remembering proposition (A) of 36, is to form the 

 Jacobians of the priminvariants with the proper quadrinvariants. 



39. First, the Jacobian of any priminvariant with a proper quadrinvarianl which 

 is itself a Jacobian is a composite function.* For, if J denote the Jacobian of P and 

 Xi (l> u we have 



But 



* This is the exact parallel of a well-known proposition in the theory of algebraical forms; see 

 CLEBSCH'S ' Theorie der binaren algebraischen Formen,' p. 117. 



