DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS 379 



by applying them to equations of the lowest orders. When they are applied to the 

 equation of the second order, they give the theorems already obtained by RUMMER 

 and SCHWARZ. When they are applied to the equation of the third order, the 

 canonical form of which is binomial and which has a single primin variant, the adjoint 

 equation is derived ; and the case of a vanishing priminvariant is discussed from two 

 points of view. The quotient-equation of the cubic, that is, the differential equation 

 satisfied by the quotient of two linearly independent solutions of the cubic, is worked 

 out ; and the primitive of the cubic is deduced from a supposed knowledge of two 

 special solutions of this quotient-equation, in a form which is the analogue of the 

 corresponding results for the quadratic. In this connexion the cubic quotient- 

 derivative occurs, corresponding to the Schwarzian derivative ; it is one of a series of 

 similar functions. For the equation of the fourth order two canonical forms are given, 

 one being the special case of the general canonical form, the other being a more direct 

 analogue of the canonical form of an algebraic binary quartic. The quotient-equation 

 is deduced and some properties are proved ; and the quartic quotient-derivative is 

 obtained. Finally, the two associate equations of the quartic are given ; and there 

 is a verification that all the priminvariants (and hence all the concomitants) of these 

 associate equations are expressible in terms of the invariants (and hence of the 

 covariants) of the quartic. 



The seventh section is really a digression from the main subject of the memoir ; 

 some of the properties of the quotient-derivatives of odd order are therein investigated, 

 the two principal relations being that which is consequent on the general quotient 

 transformation of the dependent and the independent variables, and that which gives 

 the homographic transformation of both variables. These quotient-derivatives have 

 some connexion with reciprocants ; but, on account of the restriction on the subject 

 of the memoir, there is here no investigation of that connexion. Quotient-derivatives 

 of even order are obtained from different forms of linear equations ; and a relation 

 between the two kinds of derivatives is indicated. 



The eighth and last section is mainly devoted to a proof of the functional com- 

 pleteness of the concomitants of the second, third, and fifth sections. There is a 

 homographic transformation of the independent variable, which changes one canonical 

 form into another ; and the method of infinitesimal variation is used in connexion 

 with this transformation to obtain the characteristic linear partial differential equations 

 satisfied by any concomitant. They are found to be two in number ; one of them is 

 an equation which determines the form of a concomitant, the other determines the 

 index of the concomitant when its form is known. These characteristic equations 

 are first applied to deduce the covariants which involve the original variable, 

 and next to deduce the invariants derived from 8 3 ; and simplified forms of the 

 invariants and covariants of higher grade are obtained. Finally, there is given a 

 general proof, founded on the theory of linear partial differential equations,* that 



* This method has already been applied by Mr. HAMMOND to the corresponding proposition in the 

 theory of binary quantics ; see ' Amor. Jonrn. Math.,' vol. 5, 1882, pp. 218-227. 



3 C 2 



