AND IMAGINARY ROOTS OF EQUATIONS. 
639 
colon is interposed, then it is to be understood that the facultative points lie between 
and D 2 . Thus, if we turn back for a moment to the section of G last drawn, the whole 
of the space included between the two branches and the asymptote is facultative, because 
up to the asymptote J is negative, and beyond the asymptote the whole of the space 
not included between the asymptote and the lower branch is facultative, because beyond 
the asymptote J becomes positive. Thus, then, we see that the whole of that portion of 
the plane which lies on the left-hand side of the entire curve is facultative, and the 
portion on the right-hand side of the same non-facultative ; the curve separating facul- 
tative from non-facultative space as a coast-line, indefinitely extended, separates land 
from water ; so that there is, as of course we might have anticipated, no break of conti- 
nuity in passing through the plane J. 
If we take a corresponding section of the opposite portion of space corresponding to 
the ground-plan JLFI, it is obvious that precisely the contrary takes place, because the 
sign of J is opposite in the opposite sectors ; so that what was facultative becomes non- 
facultative, and vice versd. 
(62) It is now clear that the whole of the facultative part of space is divided into 
three, and only three of the regions previously defined. One region will consist of that 
portion of it which is entirely under the plane of D : the second region will be so much 
of the upper portion as stands upon the acute sector JO A ; and the third of so much of 
the remainder of this portion as stands on the sector AOJJQn( 51 ). Again, as regards 
the second region, the line OA' is quite inoperative against its unity, because we have 
vertical ordinates above O A' through which free communication can take place between 
the blocks over JO A' and A'OA; but when we come to OA, where G touches the plane 
of D, there we have an effective line of demarcation between the adjoining blocks above 
the plane of D ; for it is impossible to pass from one into the other without going under 
D and coming up again through that plane, or else descending to the line OA and so 
meeting the plane of D ( 52 ). 
( 51 ) It will be borne in mind that the whole of the infinite prism, both above and below, standing on nOJ 
belongs to facultative space : the prism standing on the opposite section JO II, or, to speak more strictly, on the 
inside of this last-named sector, is wholly unfacultative. The facultative line D which passes through 0 is com- 
pletely isolated from the facultative portion which stands over AOJ, except at the point 0 (which we are for- 
bidden to pass through if we would remain in the same region), and is of course a rectilinear edge to the facul- 
tative prism above referred to. 
(52) Two superior regions we know d priori must exist to correspond respectively to the two cases of five and 
of one real root. Moreover we know a priori that two regions can only meet on the plane of D, and an inspec- 
tion of the dial-figure shows that only OA can be such line. Thus without completely making out the geometry 
of the question as regards the remarkable line (J=0, L=0) (the axis of D) which lies on the surface G, we may 
feel assured that the upper part of this line (which is easily found to belong to the 1 -real-root region) cannot 
have any point except the origin in common with the 5-real-roots region, since otherwise these two regions 
would communicate along this line and merge into one. When it is considered that G is a surface of the ninth 
order in J, D, L, it will not appear surprising that some difficulty arises in forming a mental conception of cer- 
tain of its local properties ; on the contrary, the subject of wonder rather is that enough can be ascertained about 
it in a very brief compass to shed all the needful light upon the analytical problem which it illustrates. 
