526 
[544 
544. 
ON A PENULTIMATE QUARTIC CURVE. 
[From the Messenger of Mathematics, vol. i. (1872), pp. 178—180.] 
I HAVE had occasion to consider with some particularity the form of a curve 
about to degenerate into a system of multiple curves ; a simple instance is a trinodal 
quartic curve about to degenerate into the form x 2 y 2 = 0, or say a “ penultimate ” of 
x 2 y 2 = 0. To fix the ideas, take x, y, z to denote the perpendiculars on the sides of an 
equilateral triangle, altitude =1 (so that x + y + z = l), and let the curve be symmetrical 
in regard to the coordinates x, y, its equation being thus 
(a, a, 1, /, f h) (x, y, zf = 0, 
where a, f h are ultimately all indefinitely small in regard to unity: to diminish the 
number of cases I further assume 
a = +, f and h = —, 
h 2 > a 2 , that is, a + h = —, 
f > a, „ ,, , V(«) H - f ' = > 
but I do not in the first instance take a, f h to be indefinitely small. Then if -/ 
is not too large, the curve is as shown in the figure 0, viz. it is a triloop curve, with 
two horizontal double tangents, 3 touching the curve in two real points, 4 touching 
it in two imaginary points. Imagine -/ increased : the new curve will have the same 
general form, intersecting the first curve at A and B but touching it at C, viz. it will 
pass inside the loop C but outside the loops A, B; and outside the remainder of the 
curve; and the 4 will also move downwards as shown. The new position of 4 will 
be below the first position. 
Supposing that a, h have given values, and that -/ continually is increased in 
regard to two things may happen. First, the double tangent 3 may move down 
1 The figure is drawn with very small values of a, f, h, in order to exhibit as nearly as may be one 
of the penultimate forms of the curve; but this is not in anywise assumed in the reasoning of the text. 
Observe in the figure that the points A, B are ordinary double points, and that there are at each of them 
two distinct tangents inclined at a small angle to each other.