638 
PEOEESSOE SYLVESTEE ON THE EEAL 
(60) Thus it will be seen that the surface G consists of two opposite portions precisely 
similar and symmetrical in respect to the axis of D. 
Let us trace that one of these whose ground-plan is comprised within the sector ITOJ. 
It will consist of two sheets coming to a cuspidal edge (a common parabola) in the 
superior part of the plane of L. The upper sheet will touch the plane of D in OA( 50 ), 
and, remaining above the plane of D, approach continually to the plane of J as an 
asymptotic plane. The lower sheet will cut the plane of D in OA', pass under the 
plane of H, cut the plane of J, progress to a maximum distance from it, and then 
approach indefinitely to J as its asymptotic plane. This will become apparent by 
taking a vertical section of this portion, cutting the lines OL, OJ; for the nature 
of the flow of the two branches of the section will evidently be as figured below, 
where/, a , X', l , K represent the points in which the lines OJ, OA', OA, OL, OJI are cut 
by the secant plane. [It should be particularly 
noticed that this figure is only intended to exhibit, 
under its most general aspect, the nature of the 
flow of the two branches of the curve ; it is drawn 
in other respects almost at random, and makes 
no pretension whatever to giving a representation 
of the actual form of the curve.] 
No part of the surface G lies under or above the 
sector nOJ, except the axis of D. The cusp C, 
where the two branches meet, is the intersection of 
the cutting plane with the parabola J=D 2 lying in 
the plane of L, and there will be another cusp at t , 
the point of maximum recession from the plane of J. 
(61) I now proceed to discriminate, by aid of 
this surface, the facultative from the non-facul- 
tative portion of space. 
If in the expression for G as a function of J, K, L we substitute for K its value 
— we G— - ^ y 4 D 4 + terms involving only lower powers of D ; so that, 
calling Dj, D 2 the two real values of D in the upper and lower sheets of G respectively 
corresponding to any point J, L, 
G=J(D-D 1 )(D-D 2 )Q, 
Q being a quantity essentially positive. 
Hence when J is negative the facultative points in any line parallel to D will be 
those for which D lies between D 1? D 2 , but when J is positive, the facultative points 
must be exterior to the segment DjD 2 ; I denote this difference in the figure by placing 
a colon between the signs in each sector for which J is positive, indicating thereby that 
the facultative points lie between -j-oo and D n and between D 2 and — oo ; but where no 
(5°) ]7 or the value of D for this sheet is zero all along OA, and positive on either side of it. 
