PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
541 
nally in terms of the coefficients of the given quintic equation and of the square root of 
the discriminant of this equation. In fact, v being arbitrary, write 
L=II 6 {v— <p(«, /3)}, M=n 6 {v— <p({3, a)}, 
then the interchange of any two roots of the quintic produces merely an interchange of 
the quantities L, M ; that is, 
L+M and (L— M) x 2 , x 3 , x 4 , x 5 ) 
are each of them unaltered by the interchange of any two roots, and are consequently 
expressible as rational functions of the coefficients ; or observing that 'Q(x l , x 2 , x 3 , x 4 , x b ) 
is a multiple of \/l ), we have L a function of the form P + Q\/D ; the equation L=0, 
the roots whereof are v=<p(u 1 , jSJ . . . v=<p(a 6 , (3 6 ), is consequently an equation of the 
form P-fQ\/.D=0, viz. it is a sextic equation (*fv, 1) 6 =0, the coefficients of which 
are functions of the form in question. Hence in particular 
u 2 = 1 2 3 45 = ( a, — x 2 ) 2 (x 2 — x 3 )\x 3 — x 4 ) 2 (x 4 — xf(x 5 — x x ) 2 
is determined as above by an equation (*$u 2 , 1) 6 =0. Another instance of such an 
equation is given by my memoir “ On a New Auxiliary Equation in the Theory of 
Equations of the Fifth Order,” Phil. Trans, t. 151 (1861), pp. 263-276. 
Article Nos. 308 to 317. — Hermite’s Canonical form of the quintic. 
308. It was remarked that M. Hermite’s investigations are conducted by means of a 
canonical form, viz. if A (=J, = No. 19 as above) be the quartinvariaut of the given 
quintic (a, b, c, d , e,ffx, yY, then he in fact finds (X, Y) linear functions of {x, y) such 
that we have 
{a, b , c, d, e,fXx, !/)‘=(b, !“■> >'XX, Y) s 
(viz. in the transformed form the two mean coefficients are equal ; this is a convenient 
assumption made in order to render the transformation completely definite, rather than 
an absolutely necessary one); and where moreover the quadricovariant (Table No. 14) of 
the transformed form is 
= n /axy, 
or, what is the same thing, the coefficients (X, p, s/k, f, X) of the transformed form 
are connected by the relations 
xf — Af/j k d - 3^~ 0, 1 
x'fA — 4:fy/k-\-3k=0, i 
XX 1 — 3 [hf -j- 2^= \/ A, j 
the advantage is a great simplicity in the forms of the several covariants, which sim- 
plicity arises in a great measure from the existence of the very simple covariant operator 
(FiL'jY ( v * z “ °P era ti n g therewith on any covariant we obtain again a covariant). 
