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

 The relation of transformation is 



t, = \^z' ]+z v = v 2'~* ! ("~ 3 > (xiv ) 



where v 3 is the corresponding transformed trilinear variable. 



And in general of the n\/p\ npl sets of ^-linear variables, each set being 

 constituted by corresponding minors of the pth order, there is only one which has 

 variables possessed of the functional invariantive property, and of this set a typical 

 variable is 



' Vi, y\, 2/1". . i/i '" 

 11 11 ' 11 " 11 (p ~ i) 



2/2> 2/2 > 2/2 > > 2/2 



<p ~ 



where as before y denotes dy/dx. If v p denote the same jp-linear variable associated 

 with the transformed equation, the law of transformation is 



+ ~ + --- 



v 



(xv.). 



The last set of variables is that for which p = n 1 ; and the typical variable of 

 the set is the variable of LAGRANGE'S " Equation adjointe." 



Associate Dependent Variables. 



59. Hence there are, in all, n 1 sets of variables ; all the variables in any one 

 set are particular and linearly independent solutions of a differential equation the 

 dependent variable of which is a typical variable of the set. Hence, connected with 

 the given differential equation, there are n 2 other differential equations ; these 

 may be called the associate equations. The n 2 new dependent variables, derived 

 by definite laws of formation, may be called the associate dependent variables , and, 

 calling them in turns the associate variables of the first, second, . . . , (n 2)th rank, 

 the differential equation of which the dependent variable is the associate of the 

 (p l)th rank is linear and of order nl/plnp\. For the functional transfor- 

 mation of the original dependent variable given by (xii.) the law of transformation of 

 the associate variable of the (p l)th rank is given by (xv.); and, if we call two 

 ranks complementary when the sum of their orders is n 2, then associate variables 

 of complementary rank are transformed by the same relation, since for such variables 

 the index of the factor power of z' has the same value. 



The associate dependent variables may therefore be ranged in pairs of complementary 



