524 
PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
ceeding in the same manner, that is, with the coefficients themselves as anxiliars, to the 
case of a quartic equation, the m-space is here a 4-dimensional space, so that we cannot 
by an actual geometrical discussion show how the 4-space is by the discriminatrix or 
hypersurface D = 0 divided into kingdoms having the characters 4 r, 2r-\-2i, M respec- 
tively. The employment therefore of the coefficients themselves as auxiliars, although 
theoretically applicable to an equation of any order whatever, can in practice be applied 
only to the cases for which a geometrical illustration is in fact unnecessary. 
267. I will consider in a different manner the case of the quartic, chiefly as an instance 
of the actual employment of a surface in the discussion of the character of an equation ; 
for in the case of a quintic the auxiliars are in the sequel selected in such manner that 
the surface breaks up into a plane and cylinder, and the discussion is in fact almost 
independent of the surface, being conducted by means of the curve (Professor Sylvester’s 
Bicorn) which is the intersection of the plane and cylinder. 
Article Nos. 268-273 . — Application to the Quartic equation. 
268. Considering then the quartic equation (a, b, c, d, e\b, 1) 4 = 0 (I retain for sym- 
metry the coefficient a, but suppose it to be =1, or at all events positive), then if I, J 
signify as usual, and if for a moment 
§=a 2 d — %abc-\-2b 3 , 
X=3aJ+2(6 2 -ac)I, 
we have identically 
f (3a 2 J 2 + X 2 )$- 2 = 9 (& 2 — acfX 2 — a 2 (b 2 —ac) 3 (I 3 — 27 J 2 )— a 2 X 3 
(see my paper, “A discussion of the Sturmian Constants for Cubic and Quartic Equations,” 
Quart. Math. Journ. t. iv. (1861) pp. 7-12). And I write 
x—b 2 —ac , 
y=2>aJ -\-2(b 2 — ac)I, 
z=I 3 — 27J 2 (=D). 
269. I borrow from Sturm’s theorem the conclusion (but nothing else than this con- 
clusion) that (x, y, z) possess the fundamental property of auxiliars (that is, that the 
quartic equations (if any) corresponding to a given system of values of ( x , y, z) have one 
and the same character). The foregoing equation gives 9 x 3 y 2 — x 3 z — y 3 = a square 
function, and therefore positive; that is, the facultative portion of space is that for 
which 9x 3 y 2 —x 3 z—y 3 is = + . And the equation 
x 3 (9y 2 -z)-y 3 =0 
is that of the bounding surface, dividing the facultative and non-facultative portions of 
space. 
270. To explain the form of the surface we may imagine the plane of xy to be that 
of the paper, and the positive direction of the axis of z to be in front of the paper. 
Taking z constant, or considering the sections by planes parallel to that of xy. 
