AND IMAGINARY ROOTS OR EQUATIONS. 
633 
(52) If, then, we insulate any portion of facultative space, and in the block so insulated 
it is possible to pass from one point to any other — that is to say, if we can draw a con- 
tinuous curve of any sort from one point to another without passing out of the block, and 
without cutting or touching the plane D=0, then by virtue of the principle just laid 
down, we see that all the points in such block have the same character, and the nature 
of the roots will be the same in the infinite number of families, each containing an 
infinite number- of individuals which the points in that block severally represent. Now 
imagine a block taken so extensive as to admit of no further augmentation, except 
accompanied with a violation of the condition of the capability of free communication 
between point and point without cutting or touching the surface D ; such a block may 
be termed a region, and the whole of facultative space will be capable of subdivision 
into a certain number of these regions. This being supposed effected, the character of 
each region will be known when we know the character of a single point in it ; that is 
to say, every region will have a determinate character of positive, negative, or neuter. 
It will presently be shown that the number of such regions is only three ( 47 ) (the least 
number it could be to meet the three cases of four, two, or no imaginary roots), one 
masculine, one feminine, one neuter; and consequently there will be but three cases to 
consider when the invariantive coordinates J, D, L are given ; according as J, D, L 
belong to one or the other of these three regions, the equation to which they belong 
will have all its roots real, or only one real, or three real and two imaginary. The 
origin, it need hardly be added, constitutes a region per se , in which, so to say, the 
characters of masculine and feminine are blended. 
(53) Let it be observed that we can see a priori that, were it not for the distinction 
between facultative and non-facultative portions of space, it would be impossible for 
each point corresponding to a given system of invariants to possess an unequivocal 
character ; for in such case there would necessarily be free continuous communication 
possible between all the points on each side of D inter se, and consequently we should 
be landed in the absurdity of conceiving the general equation of the fifth degree not 
to admit of division into cases of four, two, or no imaginary roots ; D being negative, 
we know, would imply two roots, and not more than two, being imaginary ; and accord- 
ingly D positive would imply either that four roots are imaginary or none — not sometimes 
one and sometimes the other, but in all cases alike four imaginary, to the exclusion of the 
supposition of the roots being all real, or else of all the roots being real and never four 
imaginary. Thus we see that the mere fact of a given system of invariants communi- 
cating a definite character to the roots, implies the necessity of the invariants exercising 
a restraining action over each other’s limits, and that where this restraint does not exist 
it is impossible that the character of the roots can be determined by the values of the 
invariants. 
( 47 ) It is clear from tlie definition, that a region can only he bounded by G the amphigenous surface, and D 
the plane of the discriminant : and granted (as mil he shown hereafter) that G and D touch each other in only 
one continuous line, it becomes obvious a priori that there can be but two regions on one side of D and a single 
region on the other. 
