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<^^ / Cy? ^) = ^i ^^^ therefore also aii)^ linear function of 

 those roots must satisfy the differential equation. But it is 

 known [see my Paper '^ on the Theory of Quintics," published 

 in the Quarterly Journal of Pure and Ajyplied Mathematics^ 

 Vol. III. p. 34o] that each of the constituents of the roots of 

 an equation is a linear function of those roots. Consequently 

 each of the constituents of y must satisfy the differential 

 resolvent, these constituents being in fact so many particular 

 integrals of that equation. It follows that every particular 

 integral is a linear function of the constituents, for otherwise 

 there would be more than {n — 1) independent integrals, 

 which is impossible, seeing that the resolvent equation is 

 only of the {ii — l)th order. Hence the solution of the 

 differential resolvent, that is, its complete integration so as 

 to evolve ?/, or the several constituents of ?/, in terms of x, 

 will give the required solution (algebraic, trigonometric, or 

 transcendental) of the equation in y. 



Of the two trinomial forms, suggested by Mr. Cockle, to 

 which it is known that any equation of a degree lower than 

 the sixth can be reduced by the process of Tschirnhausen or 

 Mr. Jerrard, I have selected the following 



if^ny -\-{n — \)x=0 =f{y, ^) 

 because it has, when :2;=: 1, two equal roots, and therefore 

 affords some good verifications, and also because it leads, as 

 I shall show in certain researches which I hope shortly to 

 publish in detail, to some remarkably simple expressions. 

 At present I content myself with placing on record the 

 following results : — 



In general, the first differential coefficient ~j is equal to 



— I . —1— . { y^-' + xy^^-' + - • + x^^-'y—{n~ 1) 

 n I — x"~^ { 



which gives immediately for the quadratic (n=2) the 

 differential resolvent 



2(l-x)^4-^/"l = 0...(l) 



