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

 where aj = 1 and, for values of r greater than 1, 



(a- - 1) (<r - 2)Q - 3) . . . (<r - r + l)(<r - r) 

 " r= 2.3...r(2<r-3)(2<r-4)...(2<r-r-l) 



By means of this method it would be possible to determine many, if not all, of the 

 coefficients of the general linear invariant in its uncanonical form ; this investigation, 

 however, must for the present be deferred. 



32. The general results obtained thus far may be stated as follows : 



When the linear differential equation 



d'y , r ; B _ ' p d*-'y _ 



faf "",., r In- r! r d&-'~ :a 



has its dependent variable y transformed to u by the equation 



y u\ 



and its independent variable changed from x to z, where z and A (which is a function 

 of x) are determined by the equations 



' >=-'. !=" .- 



the transformed equation in u is the canonical form 



d"u r = * nl ~. d*~ r u 



d& r ^3 rln r' dz"~ r 



The coefficients P and Q of these equations are so connected that there exist n 2 

 algebraically independent functions (#) of the coefficients P and their derivatives 

 which are such that, when the same function ,(2) is formed of the coefficients Q and 

 their derivatives, the equation 



is identically satisfied. The possible values of a- are 3, 4, 5, ...,; the function 

 X 2 ) i 8 



r = -S 



where a li(r is unity, and for the remainder of the co-efficients a 



_ (<r - 1) (<r - 2)* (ff - 3) 2 . . . (a - r + I) 8 (<r - r) 

 av ' ff ~ 2. 3 r(2<r - 3) (2<r - 4) . . . (2<r - r - 1) ' 



