412 MR. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIKNT- 



Quartinvariants. 



41. We now pass to the consideration of the invariants of the fourth degree which 

 can be obtained by the methods hitherto adopted. 



The most obvious instances of composite quartinvariants are those constituted by 

 (i) products of priminvariants by cubinvariants proper and composite, and (ii) products 

 and powers of the second degree of proper quadrinvariauts ; while proper quartin- 

 variants are to be sought among 



(a) Jacobians of priminvariants with proper cubinvariants, the Jacobians of primin- 

 variants with composite cubiuvariants being composite functions, by (A) $3G; 

 (/3) Jacobians of proper quadrinvariauts with proper quadrinvariants ; 

 (y) Quadriderivative functions of quadrinvariants, proper and composite. 



These must be considered in turn. 



42. First, for (a) ; we denote the Jacobian of a<2 and A by X,^, so that 



X ff , A = A0 X e:. s - (3o- + 3) e,, 8 el, 

 x^ = ^e, e;. 2 - (3o- + 3) e,, 8 0; ; 



and, therefore, 



xe x X,,, - fie,, x,,, = (3o- + 3) 0,, 2 e,, Mil . 



Hence it follows that, when one of the functions X, iX is known, all the other quartin- 

 variant Jacobians derived through the same cubinvariant are expressible in terms of 

 that one function and of invariants of lower degree ; and, therefore, as in 40, the 

 (n 2) 2 invariants of this type can be resolved into n 2 classes, in each of which 

 classes only one function need be retained. As before, we cboose from the class 

 derived through ^ that function which is the Jacobian of, and ffj2 ; and, denoting 

 it by ^g, we have 



,,3 = o-0, :, 2 - (3o- + 3)0 ff , 2 ;, ....... (ix.) 



a proper quartinvariant with index 4o- + 4. The number of these proper quartin- 

 variants is 71 2 ; and, in particular, the invariant ff|3 will be said to be associated 

 with the priminvariant 0^. 



When 0^3 is expressed in terms of 0, and its derivatives alone, a simpler invariant 

 can be obtained by taking a linear combination of ^3 and 0* a ; there is, however, 

 no apparent advantage at present in taking such a combination as a canonical form, 

 and there is the present disadvantage of destroying the law of formation. The modifi- 

 cations will be indicated later ( 134). 



43. Second, for (ft) ; there are three cases which occur, viz. : 



(a) The combination of a Jacobian x, M ,i with a Jacobian Q P>VI l ; 



(b) A,*I w ith a quadriderivative 0,, j ; 



(c) quadriderivative ,,! p , ^ 



