DKKIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 423 



considered in the earlier part of this memoir have possessed their invariantive pro- 

 perty for functional transformation ; and, therefore, if forms involving the dependent 

 variables are to be included in an aggregate of concomitants together with the 

 invariants, these forms must have the same invariantive property for such functional 

 transformation. In this aggregate of concomitants the variables themselves will be 

 included ; and, therefore, we must select from the foregoing algebraical combinations 

 those which have the invariantive property of reproducing themselves, save as to a 

 power of z', after transformation. 



Of the n sets of linear variables constituted by the several sets of n quantities y, 

 n quantities y", and so on, only the first set has the property of being reproduced by 

 the new variable, save as to a power of ^ ; and we already know that, if u be the new 

 dependent variable, then the relation is, by (iv.), 



y = Mz'- 4(<| -i> . . (xii.). 



Of the n(n 1) sets of bilinear variables, each set containing ^n(n 1) variables, 

 only a single one has the invariantive property of self-reproduction, save as to a 

 power of z ; and this single one is the set constituted by the n ( 1) variables of 

 the type 



This statement, which leads to the retention of the single set and the exclusion of 

 all the remainder, can be at once verified by making substitutions of the type (xii.) ; 

 and the result of the substitution on the typical variable of the present class is that, 

 if < a denote the original bilinear variable and v the transformed bilinear variable 



du. 



then we have 



dz' 



= XV v, = v,z'-<"-"= 



(xiii.). 



Similarly of the \n (n 1) (n 2) sets of trilinear variables, each set being consti- 

 tuted by corresponding minors of the third order, there is only one set of which each 

 variable has the functional invariantive property ; and a typical variable of the set to 

 be retained is 



y" yr> yr 

 y," y,', y, 



