314 



Mr. A. R. Forsyth. Invariants, &c, [Jan. 12, 



In the fifth section these dependent variables are treated in the 

 same manner as the priminvariants in the third, and give two classes of 

 functions — identical covariants, which in their canonical form, involve 

 dependent variables only, and mixed covariants, which involve depen- 

 dent variables and coefficients of the original equation. The former 

 class includes series of covariants, each involving only one of the 

 dependent variables ; the law of successive formation is 



Vp,i = p(n—p)v J ,v p " — (np—p*—l)v p '*, 



= pin—^VpY'pj—r^njp—^—^YpjVp, 



for each of the associate variables. But other functions which 

 involve more than one of the variables, e.g., the Jacobian of two of 

 them, are omitted, for they can be algebraically compounded by 

 means of the mixed covariants. The number of independent identical 

 covariants in the succession is one less than the order of the equation 

 satisfied by the variable : but a modification of this number is neces- 

 sary when they are considered as covariants of a differential quantic 

 instead of being considered covariants of a differential equation. For 

 in this case we must either retain the quantic and all derivatives 

 from it — when there is no modification of the number of identical 

 covariants ; or the number is unlimited, and then the quantic and its 

 derivatives are composite. 



The mixed covariants which are irreducible are proved to consist 

 only of first Jacobians of some one of the invariants and all the 

 dependent variables in turn. 



The aggregate of the concomitants is constituted by the three 

 classes of functions thus obtained, viz., invariants, identical covariants, 

 and mixed covariants. 



In the sixth section the results previously derived are applied to 

 equations of the second, the third, and the fourth orders ; solely, 

 however, for the sake of illustration and not for purposes of critical 

 discussion of classes of these equations. 



For the equation of the second order the only result obtained is a 

 reproduction of Schwarz's theorem; the equation has no invariant. 



For the equation of the third order, the canonical form of 

 which is 



u'" + Q 3 u = 0> 



and which has a single priminvariant, one or two questions are 

 solved; in particular, the differential equation satisfied by the 

 quotient of two solutions of the cubic is obtained, and there is thence 

 deduced a quotient-derivative, which, is the analogue of Schwarz's 

 derivative for the quadratic. 



For the equation of the fourth order there are two canonical forms, 

 viz. : — 



