405] 
AN EIGHTH MEMOIR ON Q [JAN TICS. 
157 
facultativo and non-facultative regions may be spoken of simply as the bounding 
surface, whethei the same be analytically a single surface, or consist of portions of more 
than one surface. 
202. Considei the discriminant D, and to fix the ideas let the sign be determined 
in such wise that 1) is + or — according as the number of imaginary roots is 
= 0 (mod. 4), or is = 2 (mod. 4); then expressing the equation D — 0 in terms of the 
auxiliars (A, B,... A), we have a surface, say the discriminatrix, dividing the m-space 
into regions tor which D is + , and for which I) is —, or, say, into positive and 
negative regions. 
263. A given facultative or non-facultative region may be wholly positive or wholly 
negative, or it may be intersected by the discriminatrix and thus divided into positive 
and negative regions. Hence taking account of the division by the discriminatrix, but 
attending only to the facultative regions, we have positive facultative regions and 
negative facultative regions. Now using the simple term region to denote indifferently 
a positive facultative region or a negative facultative region, it appears from the very 
notion of a region as above explained that we may pass from any point in a given 
region to any other point in the same region without traversing either the bounding 
surface or the discriminatrix ; and it follows that the equations which correspond to 
the several 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 = \ (n + 2) or 
^ (n + 1) 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) w = 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 portions of non-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 corre 
sponding to them an equation, and therefore, if the coefficients (6, 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.