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



n! 



it is finite, with value - , 1, when the associate v p is regarded as the variable 



pi n pi 



of the associate differential equation. 



The foregoing aggregate includes all proper covariants which involve v f alone in 

 their expression ; this result is derivable from the propositions which were proved 

 in Section III., and may be verified separately for the covariants. Thus, for instance, 

 if t^T denote the Jacobian of ~V pif and V ft ,, it is easy to show that 



p(n p)v f li = r(np p* 2) "V^rV^.+i s(np p* 2) V pi ,V pjr+1 , 

 whence it follows that T is composite. 



Mixed Covariants in the Original Dependent Variable. 



69. By this title invariantive functions are indicated into whose expression there 

 enter the dependent variable or variables and the coefficients of the original differential 

 equation. One method of obtaining them is that adopted in 35, viz., to combine 

 the variables and the invariants in such a form as to be absolutely invariantive, and 

 from this form derive a relative invariant which is practically a Jacobian. 



Beginning with those which involve only a single dependent variable and taking u 

 first, we have l~ l u Zir an absolute covariant, so that 



0J| - ' (z) % 2<r = 0;;- ' (a) y 2 * ; 



from which it follows, by taking logarithmic differentials, that 



0^ ' 



is a covariant of index unity, or say 



, (xxiv.) 



with index a- + 1 i (n !) 



70. The following propositions enable us to select the non-composite mixed 

 covariants : 



(i.) It is evident that, if 0, be a composite invariant, then Q, (M)J will be a composite 

 covariant ; hence we need only consider such functions as are derived from proper 

 invariants. 



With every proper invariant there is associated a proper mixed covariant of the first 

 order in u ; but, when one of these proper mixed covariants is considered as given, all 



