384 MR. A. R. FORSYTE ON INVARIANTS, CO VARIANTS, AND QUOTIENT- 



indication of his method of obtaining it ; he also gave the first of the two classes of 

 semin variants. Almost immediately after the appearance of these notes Professor 

 BRIOSCHI communicated in a letter * to M. LAGUERRE a method of obtaining the 

 invariantive results and of extending them, which, applied to the cubic and quartic, 

 led to explicit expressions for the invariants of both equations ; and the invariantive 

 property of the functions is constituted by the relation that if <j) (p, dp/dx, . . .) be 

 the function for the original equation with coefficients p, and <f> (q, dq/dz, . . .) be 

 the same function for the transformed equation with coefficients q, an equation of the 

 form 



(!)"*<* .>-*<*-> 



is satisfied. There is a premature conclusion as to the permanence of form of these 

 functions for equations of all orders, the corrected expression of which is given later 

 in the present memoir ( 28). 



6. The two notes of M. LAGUERRE and the letter of Professor BRIOSCHI are the 

 suggestive starting point of M. HALPHEN'S investigations in invariants, which occupy 

 part of his extremely valuable memoir.t So far as the invariants, qua theory of 

 forms, are concerned, the leading investigations are contained in the third chapter. 

 He there points out the functional identity of the invariants of LAGUERRE and 

 BRIOSCHI with functions previously (in July, 1878) obtained by himself |; the 

 connexion between absolute and relative invariants is derived a priori; and the 

 necessary limitation on the form of invariants arising from homogeneity in weight is 

 deduced. A method is indicated, potentially suitable for the formation of invariants, 

 by connecting the general linear equation with the linear equation of the second 

 order ; the fundamental invariant of weight 3 the same as for the cubic is derived 

 and its permanence of form for equations of all orders is pointed out ; but, except this 

 and the invariant of weight 4 for the quartic, no others are calculated. In fact, the 

 method involves extremely difficult analysis for any but the simplest cases ; and even 

 for the invariant of weight 3 an invariantive property of LAGRANGE'S " Equation 

 adjoin te" is used in addition. The rest of the memoir is devoted to the application 

 of these results. For this purpose, the author takes his general differential equation 

 in a definite canonical form so chosen that the term of order next to the highest does 

 not appear and the invariant of weight 3 is unity two relations which suffice to 

 determine the new independent variable and the multiplier of the dependent variable. 

 The applications, leading to most important deductions, chiefly concern the general 



* " Sur les Equations differentielles lineaires," ' Bulletin de la Society Mathemat. de France,' vol. 7, 

 1879, pp. 105-108. 



t " Memoire sur la reduction des Equations differeutielles lineaires anx formes integrables." ' Memoires 

 des Savants fitrangers,' vol. 28, No. 1, 301 pp. (Grand Prix des Sciences Mathe'matiques, annee 1880; 

 published 1882). 



t In his Doctor's Thesis ' Sur les invariants differentials.' Paris, 1878. 



