382 MR. A. B. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT- 

 SECT. VIII. 125, 126. Reduction of this transformation by method of infinitesimal variation. 

 (cent.) 127-129. Characteristic form-equation and index-equation satisfied by concomitants. 



130. Deductions from these characteristic equations. 



131. Application to identical covariants. 



132, 133. Simplification of "proper" identical covariants. 



134. Application to derived invariants. 



135, 136. Application to the Jacobian of a priminvariant and a derived invariant 

 associated with that priminvariant. 



137. Proof that the set of concomitants obtained in II., III., V. are functionally 



complete. 



138. General limitation in the number of identical covariants associated with a 



differential equation. 



SECTION I. 



HISTORICAL INTRODUCTION. 



1. Similarity in properties of differential equations and of algebraical equations 

 has long been of great value, both in the development of the theory and in the 

 indication of methods of practical solution of the former equations. In recent years a 

 great extension of this similarity has been made by the discovery of certain functions 

 associated with linear differential equations which are analogous to the invariants of 

 algebraical quantics ; and, principally owing to the investigations of M. HALPHEN, 

 this extension has had an important influence on the theory of cubic and quartic 

 equations and on the recognition of fresh integrable forms of equations. 



2. The most general modification of the form of an algebraical equation, without 

 causing any change in its order, is that which arises by the application of TSCHIRN- 

 HAUSEN'S transformation ; the effect of it is that, by the satisfaction of certain 

 subsidiary equations, the coefficients of terms in the transformed equation are 

 evanescent, and these terms are therefore annihilated. There exist in the trans- 

 forming relation a number of constants, taken in the first instance to be arbitrary, 

 and subsequently determined by the subsidiary equations, which, however, do not in 

 cases of high order always admit of possible algebraical solution ; and the two 

 simplest cases are those in which the transforming relation is lineo-linear and lineo- 

 quadratic. 



Now, in the case of linear differential equations, transformed without change of 

 order, there is an exact analogue of the lineo-linear relation just mentioned, whereby 

 the term involving the differential coefficient of order next to the highest is made to 

 disappear. If x and y denote the independent and the dependent variables respec- 

 tively, the relation is of the form 



y = uf(x) = u\, 



where u is a new dependent variable and X is determined by an equation of the first 



