PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
525 
2 = 0, gives y 2 (9x 3 —y)=0, viz. the section is the line y=0, or axis of x twice, and the 
cubical parabola y=x 3 . 
v 3 - 
z— +, the curve x 3 —^ ^_ ~ z has two asymptotes y—~V\\/, z •> parallel to and equidistant 
from the axis of x, and consists of a branch included between the two parallel asymptotes, 
and two other portions branches outside the asymptotes, as shown in the figure (z= +). 
2 =—, the curve 3^—^—r g has no real asymptote, and consists of a single branch, 
resembling in its appearance the cubical parabola as shown in the figure ( 2 =—). 
Taking x as constant, or considering the sections by planes parallel to that of zy , the 
equation of the section is z=9y 3 —^ which is a cubical parabola, meeting the plane of 
xy in a point on the cubical parabola y=9x 3 , and also in a twofold point on the axis of 
x, that is, touching the plane of xy at the last-mentioned point. 
271. The surface consists of a single sheet extending to infinity, the form of which is 
most easily understood by considering the sections by a system of spheres having the 
origin of coordinates for their common centre. These sections have all of them the same 
general form ; and one of them is shown (Plate XX. fig. 1), the projection being ortho- 
gonal on the plane of xy or plane of the paper, and the spherical curve being shown, the 
portion of it above the plane of the paper by a continuous line, that below it by a dotted 
line (the double point in the figure is thus of course only an apparent one) : the same 
figure shows also the sections by planes parallel to that of xy previously shown in the 
figures ( 2 =+ ) and ( 2 = — ). 
272. Now considering the discriminatrix D=0, in this case the plane 2=0, it appears 
that the bounding surface and this plane divide space into six regions, viz. above the 
plane of the paper we have the four regions, A non-facultative, B facultative, A' facul- 
tative, B' non-facultative, and below it the two regions, C facultative, C non-facultative. 
There are thus in all three facultative regions A', B, C, and since A' and B correspond 
