414 MR. A. B. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT- 



Combining these results, we see that the class (/8) furnishes no proper quart- 

 invariants. 



44. Before passing on to the class (y), it may be remarked that the results of the 

 last article are particular examples of a more general proposition, viz. : 



The Jacobian of an invariant of degree m and an invariant of degree n, where m 

 and n are greater than unity, is a composite invariant, that is, it can be 

 expressed in terms of invariants the degree of each of which is less than 

 m -}- n. 



For, calling the two invariants < and (of indices p. and v respectively), their 

 Jacobian J, and K the Jacobian of 0, and an invariant of degree n + 1 and 

 therefore of degree less than m + n we have 



J = 



K = 



and therefore 



that is, equal to the product of an invariant of degree n and an invariant of degree 

 m + 1 ; and therefore J is expressible in terms of invariants all of degree less than 

 its own. 



45. Third, for (y) ; since it follows from 34 that the quadriderivative function of a 

 composite function is itself composite, provided the proper Jacobians of composing 

 functions be considered as a prior class, we see that the quadriderivative of a 

 composite quadrin variant is a composite quartinvariant ; and, therefore, any proper 

 quartinvariants that occur in the present class will enter as either 



(a) the quadriderivative of a quadrinvariant of the type 0^ i ; or 

 (") x, Mil- 



Denoting the function in (a) by P, we have 



P = (4o- + 4) . ,6';, , - (4cr + 5) 0?, ,. 

 Also 



e,. 2 = oex, , - (20- + 2) e,, , 0:, 



0,, 3 = (T0X, 2 - (3o- + 3) 6,, , 0; ; 



and it is not difficult to prove that 



<r 2 0*P = 4(o-+ l)0,, 1 0,,a+ 4 (0- + 1) 2 0^ - (4o-+ 5)0^, 



so that P is composite, for it can be algebraically expressed in terms of invariants of 

 the first three degrees. Thus among the functions (a) there will be no proper 

 quartinvariant. 



