378 MB. A. R. FORSYTE ON INVARIANTS, CO VARIANTS, AND QUOTIENT- 



my results with those previously obtained, there is given at the end of the section a 

 very short statement of the kinds of covariantive functions which are here introduced. 



In the second section there are given the general relations between the coefficients 

 of a linear equation before and after it is subjected to the most general transforma- 

 tion. From these relations the value of the invariant 3 is deduced ; a method is 

 indicated which leads to the values of 4 , 5 , 6 , 6 7 ; and it is proved that, for the 

 first general form of differential equation adopted, there are n 2 fundamental 

 invariants, each of which consists of two parts : (i.) a part linear in the coefficients 

 and their derivatives ; (ii.) a part, not linear, every term of which contains at least 

 one factor which is either the algebraic coefficient of the term next but one below the 

 highest in order in the differential equation, or is a derivative of that algebraic 

 coefficient. A canonical form of the differential equation is adopted, the reduction to 

 which is possible by the solution of an equation of the second order ; for this canonical 

 form the second part of each of the fundamental invariants vanishes. Finally, the 

 expression of these invariants in their canonical form is given. 



In the third section two processes of deducing invariants from those already found 

 are obtained, called the quadriderivative and the Jacobian ; and it is proved that all 

 the algebraically independent invariants which can be deduced by these processes may 

 be arranged in classes according to their degrees in the coefficients of the differential 

 equation. The first class is constituted by the n 2 priminvariants of the second 

 section ; the second class contains n 2 quadriderivatives of these priminvariants 

 and n 3 independent Jacobians ; and each succeeding class contains n 2 proper 

 invariants. In the course of the section several propositions are proved which lead to 

 this selection of proper invariants. 



In the fourth section it is shown, by the application of CLEBSCH'S theorems as to 

 the classes of variables which arise in connexion with the concomitants of algebraical 

 quantics in any number of variables, that there are in all n 2 dependent variables, 

 associated with the original dependent variable of the differential equation, and 

 distinct in character from one another. The complete set of n 1 dependent 

 variables are subject to similar linear transformations ; and at the end of the section 

 some properties of the linear equations satisfied by them are inferred. 



In the fifth section the quadriderivative and Jacobian processes are applied to the 

 dependent variables, original and associate, which possess the invariantive property ; 

 and it is proved that there are two classes of independent co variants, viz., those 

 which involve each one dependent variable and its derivatives only, and those which 

 are Jacobians of a single invariant and each of the dependent variables in turn. A 

 limitation on the former class, according as they are considered associated with a 

 differential equation or a differential quantic, is pointed out ; and a symbolical 

 differential expression is obtained for each of the proper derived invariants and 

 derived covariants. 



In the sixth section some illustrations of the theorems already proved are given, 



