DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 427 

 where 



y,. y. y*> y. 

 tf?> y* yV> y*. 

 y",. y",. A y. 



If y^ 77 f . v 



and similarly for v^ ; hence it is expressible in terms of associates of the original 

 variable y. 



As a last example, consider the set of variables associate of the first rank with t a ; 

 one of these variables may with the foregoing notation be written 



v m*m-*rn v * 



and this easily proved to be 



again expressible in terms of associates of the variable y. 



62. From these particular results and the preceding investigations the following 

 inferences may be drawn : 



(1.) The system of associate variables, constituted by y, t z , t z , . . . , f_i, is 

 functionally complete ; that is to say, the variables in the systems associate with 

 any one of them (derived from that one by the functional operations of the type 

 which led to y, t^ t 3 , . . . ) are, qua variables, expressible in terms of combinations of 

 particular associates of the original variable y ; in the formation of these combinations 

 it may be necessary to introduce functions of the coefficients of the original equation. 

 Hence, as typical dependent variables, the associates of the variables associate with y 

 may be looked upon as expressible in terms of the variables associate with y, the 

 necessary combinations of which are only multiplicative and additive ; and they 

 therefore introduce no new associate variables. 



(2.) Invariants of associate equations are all of them invariants of the original 

 equation ; the complete converse of this may not be affirmed. 



(3.) Differential equations in associate variables of complementary rank are 

 mutually " adjoint." 



The last inference is suggested by the following considerations : When we 

 construct the equation adjoint to the differential equation, of which the dependent 

 variable is t p and order n \jp 1 n p !, the process can be performed in a manner 

 similar to LAGRANGE'S adopted in 52. The integrating factor, which is the 

 dependent variable of the adjoint equation required, can be constructed as a 

 functional determinant of special solutions t of the equation, in number one less than 

 the order of the equation. For the special integrating factor, corresponding to that 

 of 52, let the particular t solution omitted from the determinantal expression be the 

 functional determinant of y l9 y& . . . , y r When, in the integrating factor, determinant 

 substitution takes place for the particular solutions in terms of the quantities y, it 

 appears that the determinant is of dimensions 



3 I 2 



