PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
■523 
points of the same region have each of them one and the same character ; that is, to a 
given region there correspond equations of a given character. 
264. It is proper to remark that there may very well be two or more regions 
which have corresponding to them equations with the same character; any such 
regions may be associated together and considered as forming a kingdom ; the number 
of kingdoms is of course equal to the number of characters, viz. it is 2) or 
according as n is even or odd; and this being so, the general conclusion from 
the preceding considerations is that the whole of facultative space will be divided into 
kingdoms, such that to a given kingdom there correspond equations having a given 
character; and conversely, that the equations with a given character correspond to a 
given kingdom. Hence (the characters for the several kingdoms being ascertained) 
knowing in what kingdom is situate a point (A, B, . . . K), we know also the character of 
the corresponding equations. 
265. Any conditions which determine in what kingdom is situate the point (A, B, . . . K) 
which belongs to a given equation (1 ,b,c . . l) ra = 0, determine therefore the character 
of the equation. It is very important to notice that the form of these conditions is to a 
certain extent indeterminate ; for if to a given kingdom we attach any portion or por- 
tions of n on-facultative space, then any condition or conditions which confine the point 
(A, B, . . . K) to the resulting aggregate portion of space, in effect confine it to the 
kingdom in question ; for of the points within the aggregate portion of space it is only 
those within the kingdom which have corresponding to them an equation, and therefore, 
if the coefficients ( b , c , . . .) of the given equation are such as to give to the auxiliars 
(A, B, . . . K) values which correspond to a point situate within the above-mentioned 
aggregate portion of space, such point will of necessity be within the kingdom. 
266. In the case where the auxiliars are the coefficients ( b , c , . . .), to any given values 
of the auxiliars there corresponds an equation, that is, all space is facultative space. 
And the division into regions or kingdoms is effected by means of the discriminatrix, or 
surface D = 0, alone. Thus in the case of the quadric equation (1, x, y^J, 1) 2 =0 the 
W 2 -space is the plane. We have D=^ 2 — y, and the discriminatrix is thus the para- 
bola x 2 —y= 0. There are two kingdoms, each consisting of a single region, viz. the 
positive kingdom or region (x 2 — y=-\-) outside the parabola, and the negative kingdom 
or region (x 2 — y=~) inside the parabola, which have the characters 2r and 2 i, or corre- 
spond to the cases of two real roots and two imaginary roots, respectively. And the 
like as regards the cubic (1, x , y , zj{6, 1) 3 =0; the m-space is here ordinary space, 
D= — \.x 2 z-\-Sx 2 y 2 -\-§xyz — iy 2 — -z 2 , and the division into kingdoms is effected by means 
of the surface D = 0 ; but as in this case there are only the two characters Sr and r-\-2i, 
there can be only the two kingdoms D= + and D=— having these characters Sr and 
r-\-2i respectively, and the determination of the character of the cubic equation is thus 
effected without its being necessary to proceed further, or inquire as to the form or 
number of the regions determined by the surface D = 0: I believe that there are only 
two regions, so that in this case also each kingdom consists of a single region. But pro- 
4 b 2 
