AND IMAGINARY ROOTS OR EQUATIONS. 
665 
Description of the Plates. 
PLATE XXIV. 
The ( g , rf) equation is (1, g, g 2 , rf, rj, l^jv, y) 5 =0, of which two roots are always imagi- 
nary; its extreme criteria are 0, 0; its middle criteria s 4 — s/i 2 , rf — riz 2 , 
p=&ri—l, <r=(s 3 — rf)(i 2 — rf)‘ 
Points (p, a) above the discriminatrix indicate 2 pairs of associated roots in the (g, n) 
equation. 
Points (p, <r) on the discriminatrix indicate 2 equal roots in the (g, rf) equation. 
Points ( p , a) under the discriminatrix indicate 3 solitary roots in the (s, rf) equation. 
Points (p, <r) above the equatrix indicate s, n real and unequal. 
Points ( p , <r) on the equatrix indicate g, rj equal. 
Points ( p , a) under the equatrix indicate g, n imaginary and conjugate. 
Points ( p , <r) above the loop of the indicatrix indicate middle criteria not both positive. 
Points ( p , a ) on the loop of the indicatrix indicate middle criteria of opposite signs. 
Points ( p , a) under the loop of the indicatrix indicate middle criteria not both negative. . 
The discriminatrix is a closed curve, the whole of which is figured on the Plate, and 
is shaped somewhat like a harp : it has a cusp of the fourth order at the origin. 
The equatrix consists of two branches coming together at a cusp at the distance 1 
from the origin ; the upper branch touches the horizontal axis at the origin ; the lower 
branch, after touching the discriminant at a single point, sweeps out from and round it, 
cutting the vertical axis at the distance 4 below the origin. Both branches go off to 
infinity to the right, and lie completely under the horizontal axis. Where the lower 
branch touches the discriminatrix, the discriminant of the (g, rf) equation passes through 
zero without changing its sign. 
The indicatrix consists of a single branch extending indefinitely in both directions. 
It passes from infinity below and to the left until, at the distance 1 from the origin, it 
touches the axis, which at the origin it crosses at an angle of 45°, after which it goes off 
to infinity in the positive direction. Its loop extends from p = () to p— — 1. The two 
portions of it figured in the Plate join on together, coming to a maximum at a great 
distance below the horizontal axis. The narrow tract marked “ Legion of Real para- 
meters ” is that portion of the harp-shaped space for which alone, g, n being real, the 
(g, rf) equation can have more than one real root. The areas of each of the three regions 
into which the discriminatrix is divided by the equatrix and indicatrix may readily be 
expressed numerically in terms of algebraic and inverse circular functions only. 
I am indebted to Gentleman Cadet S. L. Jacob, of the Royal Military Academy, for 
the tracing of the curves of which the above Plate is a somewhat imperfect reproduction. 
PLATE XXV. 
Described in text, p. 658. 
