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



as can be seen at once by multiplying the foregoing equation by v and integrating by 

 parts. This is LAGRANGE'S associate equation (" Equation adjointe ") ; it is of the 

 same order as the original equation ; and its special and independent integrals may be 

 taken to be the n determinants each of (n I) 3 constituents given by 



(n-2) (n-2) (n - 



y. , y._i , ys 



(n-3) fn-3 (it- 



y/ }/ -J/ 



y , y.i-\ , ?/*-2 



(n-2) 

 /I 

 (n-J) 



It is well-known that the Lagrangian associate of the Lagrangian associate is the 

 original equation ; it is evident that, if either be in its canonical form, the other is so 

 also. It will now appear that the equation is only one of a set of equations, and its 

 variable only one of a set of dependent variables, associated with the original equation. 



Sets of Variables subject to the. same Linear Transformation; Algebraical Combination. 



53. The n special integrals y constitute a fundamental system of integrals, and 

 each of the members Y 1 , Y 2 , . . . , Y, of any other fundamental system is a linear 

 function of the former set, so that in effect a change from one fundamental system 

 to another is only a linear transformation of the dependent variables concerned. 

 (There is here no question of the necessary modifications of fundamental systems 

 owing to the presence of "singular" values of the independent variable). This 

 transformation may be represented by 



!,...,.) = (MX ??* 



where M is a constant matrix with a non-vanishing determinant. But this applies 

 not only to the dependent variables, but also to their derivatives of all orders, so that 

 we have 



x (tSu (tyf I \ w*v dx rt* 



for any value of r. And, if we retain this equation for values 0, 1, 2, ...,. I of 

 the index r, we shall have in all n sets of variables subject to the same linear trans- 

 formation ; and these variables are linearly independent of one another, since for the 

 satisfaction of the differential equation we need the nth differential coefficients of the 

 quantities y, which have been specially excluded. 



