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XV. Invariants, Covariants, and Quotient- Derivatives associated with Linear 



Differential Equations. 



By A. R. FORSYTH, M.A., F.R.S., Fellow of Trinity Colleyp, Cambridge. 



Received January 7, Read January 12, 1888. 



THE present Memoir deals with a set of invariants and covariants of linear differential 

 equations of general order. The set is proved to be complete, that is to say, every 

 covariantive function of the same type can be expressed as a function of the members 

 of the set, the only operations necessary for this expression being purely algebraical 

 operations. The transformations, to which the differential equations are subjected, 

 are supposed to be the most general consistent with the maintenance of their order 

 and their linear character ; they are, linear transformation of the dependent variable 

 and arbitrary transformations of the independent variable. The covariantive property 

 of the functions considered is constituted by the condition that, when the same 

 functions are formed for the transformed equation, they are equal to the functions for 

 the original equation, save as to a factor of the form (dz/dxy, where z and x are the 

 two independent variables. 



The memoir, with the exception of a single and rather important digression, is 

 occupied solely with investigations of the forms of the functions, of their interdepen- 

 dence, and of methods of construction. The earlier part deals chiefly with the 

 synthetic derivation of the functions, the later part with their analytic derivation. 

 Tables of the functions have not been calculated ; in most cases the expressions of 

 the functions are given in their forms as associated with the differential equation 

 when it is taken in an implicitly general canonical form, and only in very few cases 

 are functions given in connexion with an explicitly general form. Within these 

 limits the subject of the memoir has been strictly confined ; there is not, for instance, 

 any attempt at classification of differential equations of the same order as discrimi- 

 nated by forms and values of invariants or covariants. 



The contents of the memoir are as follows : 



The first section gives references to previous writers on the subject, viz., COCKLE, 

 LAGUERRE, BRIOSCHI, MALET, and HALPHEN ; and, in particular, some of the results 

 obtained by HALPHEN in his well-known essay and in a subsequent memoir are stated. 

 It appears that previous results are confined to invariants, and that, with the 

 exception of two special invariants of the general equation, the invariants obtained 

 are not derived for equations of order higher than the fourth. In order to connect 



MDCCCLXXXVm. A. 3 C 9.11.88 



