DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 383 



order. The analogue of the lineo-quadratic relatiou apparently does not exist ; but 

 another equally effective transformation of the differential equation is possible, being 

 that whereby the independent variable is changed. And, by a proper transformation 

 to a new variable z, concurrently with the former change of the dependent variable, 

 though with a different multiplier f (x), it is possible to remove the terms which 

 involve the two differential coefficients of order next below the highest which occurs. 

 This has been known for some time, having been pointed out, first apparently (in 

 1876) by COCKLE, and afterwards (in 1879), independently, by LAQUERRE. 



3. Here would seem to be the limit in this regard to the analogy between alge- 

 braical and differential equations ; but within the limit there are striking properties 

 in common. It is well known that when the proper lineo-linear transformation is 

 applied to an algebraical equation so as to remove the term next to the highest, the 

 remaining coefficients are the algebraical coefficients of the leading terms of HERMITE'S 

 covariants associated with the quantic which is the sinister of the equation ; and 

 these coefficients are therefore seminvariants. An exactly similar property holds for 

 differential equations, but its full recognition has only been gradual The following 

 are, so far as I can discover, the chief references to this part of the subject, and, 

 though a chronological order is avoided, they will serve to indicate the development. 



4. In a memoir entitled " On a Class of Invariants," * Professor MALET obtained, 

 and applied to the solution of special questions connected with the cubic and quartic, 

 two classes of seminvariants of differential equations ; one of these is invariantive 

 for change of the dependent variable, the other for change of the independent 

 variable. And though, to obtain the form of the latter he has used the two kinds of 

 transformation successively, he has not apparently obtained in a direct form functions 

 which possess the invariantive property for both transformations. Soon after the 

 appearance of this paper, and in connexion with it, Mr. HARLEY t proved that 

 Professor MALET had been anticipated by Sir JAMES COCKLE, who had in several 

 memoirs (exact references are given by Mr. HARLEY) given in forms, sometimes 

 explicit and sometimes implicit, the leading results obtained by Professor MALET 

 relating to the seminvariants of the two classes. At the end of his paper Mr. HARLEY 

 states that, in a recent letter, Sir JAMES COCKLE had suggested the possibility of 

 forming " ultra-critical" functions, i.e., functions invariantive for both transformations 

 effected concurrently. 



5. In this last suggestion, which is not stated to have been worked out to a definite 

 issue, Sir JAMBS COCKLE has been anticipated by M. LAOUERRE, who in two notes J 

 gave what is here called the fundamental invariant of the cubic, but without any 



* Phil Trans.,' 1882, pp. 751-776. 



t " Professor MALET'S Classes of Invariants identified with Sir JAMES COCKLE'S CriticoidB," 'Roy. Soc. 

 Proc.,' vol. 38, 1884, pp. 44-57. 



J " Sur les equations differentielles lineaires du troisieme ordre," ' Comptes Rendns,' vol. 88, 1879, 

 pp. 116-119 : " Sur quelqnes invariants des equations differentielles lineaires," ibid., pp. 224-227. 



