AND IMAGINARY ROOTS OF EQUATIONS. 
629 
reduced form au 5 -\-5 bidv +10 V, that on such hypothesis all the invariants J, K, L 
must vanish, so that JK 4 is still non-negative ( 40 ). 
(44) It is most important to notice that G can only become zero by virtue of two of 
the quantities g, <7, r, and therefore of r, s, t becoming equal. When u, v are imaginary, 
it is the coefficients r, s which must become equal, as otherwise the reduced form would 
not be a real function of x, y. By equating r to s, and using as an auxiliary variable 
the ratio y or y, we shall be able to study the composition and inward nature of G with 
the utmost clearness and facility. 
Section II. — On the Criteria which decide the Number of Meal and Imaginary Moots. 
(45) Since in the preceding section we have supposed that u, v are always real linear 
functions of x, y, it is obvious that the character of the roots of the given quintic in x, y 
is completely identical with that of the roots in the reduced form, and it has been shown 
that only one reduced form corresponds to a given system of values of J, D, L( 41 ). 
Let us suppose J, D, L to be taken as coordinates of a point in space ; when J, D, L 
are so related that the condition G non-negative is satisfied, the point will correspond 
to an equation with real coefficients, and may be termed a facultative point. But 
when G is negative it will correspond to an equation of the kind alluded to in the 
recent section of this paper, and there called conjugate : such a point may be termed 
non-facultative. Thus the whole of space will be divided into two parts, separated by 
the surface G=0, which may be termed respectively facultative and non-facultative (as 
being made up of facultative or non-facultative points ( 42 ) ). It is clear that these two 
portions will be exactly equal, similar, and symmetrical with regard to the axis of D ; 
by which I mean that, if two points be taken in any line perpendicular to the axis of D 
at equal distances from that axis, one will be facultative and the other non-facultative, 
as is evident from the fact that when J, L become — J, — L (K, and therefore D or 
J 2 — 128K, remaining unaltered), G is converted into — G. Thus by a semirevolution 
(4°) \yhen the form is au 5 -f 5euv 4 +fv s so that L=0, the canonizant, as has been seen before, is ae~v 2 u; the 
resultant of these two is a s e w a?f=a 7 e 10 f. Again, J=<r/ 2 , K= —2a 3 e 5 ; thus the square of the resultant = 1 LJK 4 ; 
so that if we call this resultant, which we may take as the definition of the Octodecimal Invariant I, we have 
G=16I 2 . 
( 41 ) It should be well noticed that the mere ratios ^ ^ do not suffice to determine the character of the roots. 
When these ratios are given, it is true that the ratios r, s, t in the reduced form are given, but according as L 
is positive or negative, the arguments u, v in ru s -f- sv 5 -f tiv s (supposing w to be the real linear function of x, y) 
will be real or imaginary. When J, L, D are all given absolutely, then the character of the roots is completely 
determined. The indelible marks of a quintic function are three in number, viz. the ratios ~ , ~ and the sign 
O 
s 3 
of L or J, as for a quartic function they are two in number, viz. — and the sign of s. 
( 42 ) It will also be convenient to call the coordinates J, D, L corresponding to any facultative point a facul- 
tative system of invariants, and 5 corresponding to the same (for a given sign of J) a facultative system of 
J" J* 5 
invariantive ratios. 
4 p 2 
