DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 385 



cubic and a limited form of the quartic for which the invariant of weight 3 vanishes 

 identically, and in the case of which the new independent variable is determined by 

 taking the fundamental invariant of weight 4 to be unity ; in the notation of this 

 memoir such a quartic would assume the binomial form 



7. In a subsequent memoir,* M. HALPHEN considers the general quartic and its 

 invariants, which he identifies (p. 330, I.e.) with the differential invariants of tortuous 

 curves. And he deduces (p. 339, I.e.) from the two fundamental invariants 

 (v, 7 = B 3 , 8 4 , in my notation to a numerical factor pres) the series of successive 

 invariants, which are the successive " Jacobian derivatives " herein obtained. 



The following investigations were completed before I knew any of the details of 

 this last- quoted memoir by M. HALPHEN, my starting point having been M. BRIOSCHI'S 

 letter ; and, though the results relating to the form of the Jacobian series for the 

 quartic are thus anticipated by four years, it does not seem necessary to modify the 

 investigations which relate to the equation of general order n possessing n 2 

 fundamental invariants. 



8. The great advantage of the canonical form chosen by M. HALPHEN is that a 

 given equation can be reduced to it by means of differential equations of soluble 

 form of the first order only that is, their dependent variables can be explicitly deter- 

 mined as functions of the independent variable, though the functions may not be 

 evaluable in known forms ; but there is an attendant disadvantage from the point of 

 view of the invariants that their expressions, even for the canonical form, remain 

 complicated. In preference to M. HALPHEN'S canonical form I choose that from 

 which the two terms of order next to the highest are absent, and the reduction to 

 which is always possible by the solution of a linear differential equation of the second 

 order. The great advantage of this, as the canonical form, is that, when the invariants 

 at first called fundamental and subsequently priminvariants on account of the 

 property about to be mentioned are constructed for this form, they are purely linear 

 functions of its coefficients and their derivatives, with the further essential property 

 that the expression of each is independent of the order of the equation, so that, in 

 fact, each is an invariant of every equation of order not less than its index. 



9. The number of these priminvariants is n 2 ; from them there are constructed 

 the series of what have been called derived invariants, which include Jacobian and 

 quadriderivative functions ; and in the aggregate only those are retained which are 

 proper or non-composite. All these functions are entitled invariants. 



There is then indicated a set of dependent variables associated with the dependent 

 variable of the given equation, the last one of which set is the variable in the 



" Snr les invariants des equations diflterentiellea linruircs du qnatri&me ordre." ' Acta Math.,' vol. 3, 

 1883. pp. 325-380. 



MDCCCLXXXVIir. A. 3 D 



