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of f (y, x) — 0, and therefore also any linear function of 
those roots must satisfy the differential equation. But it is 
known [see my Paper “ on the Theory of Qiiintlcs,” published 
in the Quarterly Journal of Pure and Applied Mathematics , 
Vol. III. p. 348] 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 ( n — 1 ) th order. Hence the solution of the 
differential resolvent, that is, its complete integration so as 
to evolve y , or the several constituents of y, 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 
V n — ny + (n — 1) x = 0 —f (y, x) 
because it has, when x = 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 -f is equal to 
S.X 
1 
n 
1 
y n ~ x + rij- + • • • + x y - 
■ (n— 1) j; 
1 —x"- 1 
which gives immediately for the quadratic (n — 2) the 
differential resolvent 
2 ( 1 — x) + y — 1 = 0 (1) 
