DKIUYATIVKS ASSOCIATED WITH LINEAR DIFFKUKNTIAL EQUATIONS. 421 



54. Since the n quantities y are linearly independent of one another, they may 

 be looked upon as the coordinates of a point in a manifoldness of n I dimensions ; 

 and, if we assume the same linear independence of the derivatives of all the orders 

 up to the (n l)th inclusive (which is equivalent to an assumption that no linear 

 function of the quantities y with constant coefficients is equal to a rational integral 

 algebraical function of order less than n I an assumption justifiable with general 

 coefficients, though not necessarily so in any particular case), then each of the n 1 

 sets of derivatives, each set being constituted by those of the same order, may be 

 looked upon as the coordinates of a point in a manifoldness of n 1 dimensions. 

 And, since the law of linear transformation is the same for all the sets, all these 

 points may be taken as belonging to the same manifoldness. There are thus n different 

 and independent sets of cogredient variables connected with the single manifoldness 

 of n 1 dimensions. 



55. In the theory of the concomitants of algebraical quantics of any order in the 

 variables of a manifoldness of n 1 dimensions, it is necessary to consider all tho 

 possible classes of variables which can enter into the expressions of these con- 

 comitants. CLEBSCH * has proved that there are in all n 1 different classes of 

 variables which thus need to be considered, and that, if x s ..... x, ; y lt y z , . . . , y,; 

 Zj, z 2 , . . . , 2, ; ... be n sets of cogredient variables, the several classes are constituted 

 by minors of varying orders of the determinant (itself an identical covariant) 



Z 2 , . . . , Z, 



those of one class being minors of one and the same order. The variables of any class 

 are linearly, but not algebraically, independent of one another, except in the case of 

 the first class, constituted by minors of order unity, and the last class, constituted by 

 minors of order n 1 (the complementaries of those of the first class), in each of which 

 classes the n variables are quite independent of one another. And all similar combina- 

 tions of variables are expressible in terms of variables actually included in the classes. 

 56. In connexion with our differential equation we have obtained n different and 

 algebraically independent sets of cogredient variables ; the functional derivation of 

 the sets, one from another in succession, by the process of differentiation has been 

 excluded from any interference with their algebraical independence. We already 

 have one class of variables, viz., y lt y. 2) . . ., y, analogous to the first class of algebraical 

 variables, and another class of variables, viz.,v l( v 2 ,. . . , v m analogous to the (n l)th 

 class of algebraical variables ; and the relation 



* " Ueber cine FundamentalaufRabe der Invariantentheorie," ' Gottingen, Abhandlungun,' vol. 17, 1872. 



