1888.] 



Linear Differential Equations. 



311 



II. " Invariants, Covariants, and Quotient Derivatives associated 

 with Linear Differential Equations." By A. R. Forsyth, 

 M.A., F.R.S., Fellow of Trinity College, Cambridge. Re- 

 ceived January 7, 1888. 



(Abstract.) 



The present memoir deals with, the covariantive forms associated 

 with, the general ordinary linear differential equation ; it is strictly 

 limited to the consideration of those forms, without any discussion of 

 their critical character. 



The most general transformation, to which such equation can be 

 subjected without change of linearity or of order, is one whereby the 

 dependent variable y is transformed to u by a relation 



y = uf(x), 



and, at the same time, the independent variable x is changed to z ; 

 and. when these transformations are effected on 



i n P n 1 dn ' r y - o 



r=0 



so that it becomes 



= n „ n ! d n ~ r u A 

 r ! n—r ! clz n r 



r=0 



there are r relations between the coefficients P and Q ; and P and 

 Q may manifestly, without loss of generality, be taken equal to 

 unity. 



It is shown that, from these relations, others can be deduced which 

 are of the type 



IK?) = (!)V(Q). • 



where ^-(P) is an algebraical function of the coefficient P and their 

 derivatives. Such a function is called an invariant of index p ; and 

 irreducible invariants are of two classes, fundamental and derived. 



It is convenient to have expressions for the invariants, when the 

 differential equation has an implicitly general canonical form. In the 

 first place it may be supposed that P 2 and Q x are both zero ; otherwise 

 both equations can by substitutions of Y for y 8-/*** and U for 

 ue/QJz \)q reduced to forms in which the terms involving the (n— l)th 

 differential coefficients of the dependent variable do not occur. The 

 relation between the dependent variables is now 



