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



which is satisfied, is precisely the same as the corresponding relation between the 

 similar variables helping to define the higher class (CLEBSCH, 1. c., p. 4). Hence, 

 from the point of view of purely algebraical forms, we infer that the suitable 

 algebraical combinations of the sets of variables, which have arisen in connexion with 

 the differential equation, are the minors of varying orders of the determinant 



A = 



!/i .2/2 . 



y\ 



y,, 



(n-D 



2/2 , 



y* 



which determinant, as we know, is a non-evanescent constant ; and these variables 

 may be ranged in classes, which for the present may be called linear, bilinear, tri- 

 linear, .... 



Algebraical Combinations functionally Invariantive. 



57. Now, after having obtained the merely algebraical result, it is necessary to take 

 account of functional dependence of the sets due to differential derivation. In the 

 case of the algebraical quantics, it is a matter of indifference which set of minors of 

 the first order be taken to constitute the first class of variables, which set of minors of 

 the second order be taken to constitute the second class, and so on ; thus for the 

 second class the same kind of variable is obtained by taking the (xy] minors, as by 

 taking the (yz) minors, or the (xz) minors. But a difference arises in the case of the 

 variables occurring in connexion with the differential equation. There are n sets of 

 linear variables distinct in character from one another ; for y\, i/ 2 , . . . , y' n are special 

 integrals of an equation quite different from the original equation, though they are 

 subject to the same law of linear transformation as y lt y 2 , . . . , y H . There are 

 ^n (n 1) sets of bilinear variables distinct in character ; thus 



y i y\ 



y i. y\ 



y i. y\ 



are three distinct variables of this class, subject to the same law of linear transforma- 

 tion ; and so on for the higher classes. 



58. Most of these, however, will be excluded. These forms of variables have been 

 suggested in connexion with the theory of linear transformations, for which trans- 

 formations algebraical concomitants involving them are covariantive. The invariants 



