I>i;ilIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 387 



SECTION II. 



PRIMINVARIANTS OP A LINEAR DIFFERENTIAL EQUATION. 

 Transformation of the Differential Equating 



11. The general linear differential equation of the 71 th order is of the form 



rf-Y rf-'Y (-!) rf-Y 



+ ~T ~ * 



where RQ, R lf R 2 , . . . are functions of x ; by the substitution 



and subsequent division throughout by R , it is changed to 



where P 2 , P 3 , . . . are seminvariants of the former equation. 



Similarly, an equation which determines another dependent variable u as a function 

 of another independent variable z may be written in the form 



n\ d*~*u nl 



Suppose now that these dependent variables are so connected that the relation 



y = x . . ......... (i) 



is satisfied, X being some function of x. In order that (L) may be transformable into 

 (ii.), z must be some function of x; and when this is the case there will be a number of 

 equations, evidently u in number, connecting X, z, x, and the two sets of coefficients 

 P and Q, which may be obtained as follows. The actual substitution of X for y 

 in (1) gives 



rfM dX (fr-Ht n! <iPX (fr-*u 



= dj? + " dx <k- f "*~ 2 ! n - 2 ! rfx <**- T. ', ! .' 



"- , tt rfX rf- 



- x + ^- 2) rf 



3 D 2 



