AND IMAGINARY EOOTS OF EQUATIONS. 
663 
the axis of y for its asymptote, lies entirely in the — y, —z quadrant, and is always 
convex to the axis of y; the second branch, joining the first at a cusp corresponding to 
cp= — -f, is concave to the origin, cuts the axis of y negatively and of z positively, and 
goes off to infinity ; the third branch, having the positive part of the axis of y for its 
asymptote, lies in the +y, -\-z quadrant, is always convex to the axis of z, which it 
touches at a point below that where it is cut by the second branch, and also goes off 
to infinity, lying entirely under the second branch. A straight line, according to the 
direction in which it is drawn, may cut the curve in one, three, or five real points. 
(98) When D is constant, writing J=z, L=x, we have 
D 
P 8 (P + i) 
(? + 2)*(f>— 3)’ 
Dz 
+ 2 ) 2 ' 
The form of the curve changes with the sign of D. 
the plane of D, we may make 
D=c 2 , T 2 =P±i, or <p= 
®-3’ r 
For sections parallel to and above 
3t 2 + 1. 
T 2 — 1 ’ 
then the complete equation-system to the curve will be 
( t 2 -!) 4 
3t 2 +1 
5t 2 — 1 
X — &T 
4(5t 2 — 1) 
it being unnecessary to affect c with a double sign, since z and x change their signs with 
that of t. 
Also 
lx (t 2 + 1)(15t 2 + 1)8t 
x t(t 2 — 1)(5t 2 — 1) 5 
V c 3 (t 2 +1)(15t 2 + 1)(t 2 -1)\ 
° X — 4 (5t 2 — ] ) 4 ' 
dx c 2 (t 2 + l)(r 2 — ]) 2 
Tz~ 4 (5t 2 — l) 2 
Iz 
(t 2 — 1)(15t 2 + 1)St 
t(3t 2 + 1)(5t 2 — 1) ’ 
(I5r 2 + 1)(t 2 — 1) 
St, 
To the values of r included between +\/i a »d —\/\ will correspond one branch of 
the curve passing through the origin, where it has a point of contrary flexure, and 
extending to infinity in both directions. 
When (5 r 2 — 1) is positive — is always positive; and when r 2 =l, 
&p=0, Zz= 0, |=0. 
Hence there will be a cusp of the second kind when a?=0, z=+c, the axis of z being 
a tangent to the curve at each cusp. One pair of branches has its cusp at the point 
a;=0, z—c, and the values of x, z increase indefinitely in the respective branches as r 
passes from 1 to -foo and from 1 to \/\. This pair lies in the -\-x , -j-z quadrant, and 
there will be a precisely similar and similarly situated pair in the — x, —z quadrant. 
Thus there will be in all one infinite J- formed branch passing through the origin, and 
