104 
Proceedings of the Royal Society of Edinburgh. [Sess. 
This equation leads in general to a cubic in t, with at least one real 
root ; A and B may then be chosen in an infinity of ways. 
When = one rea q roo t j s t = 0; and when ^- 2 — — a rea J roo t 
b 0 n 0 b 2 n x 
is t = oo . 
If the numerator and denominator of the left side of (7) have a common 
factor t — a, then, for t = a, (1) is already the line at infinity, and its envelope 
is a parabola. 
The case in which the numerators of (7) or the denominators of (7) have 
a common factor presents no difficulty. 
When the numerator and denominator of the right side of (7) have a 
common factor, the pencil of lines consists of a system of parallel lines with 
the vertex at infinity. 
In such a case a change of parameter and change of axes will enable us 
to write (1) and (2) as 
L 0 + L-p + L 2 t 2 = 0 (8) 
x + t = 0 . . . . . (9) 
and the equation (7) as 
a 0 + a 1 t + a 2 t 2 _ 1 
b 0 + bpt + b 2 t 2 0 
Hence 
b^ + bp; + b 2 t 2 = 0 ...... (10) 
so that t and therefore A and B may not be real or may be real. Thus, 
when the double point is at infinity, the parabolic envelope may not or may 
be real. 
In any case, the analysis leads to the conclusion that, when the double 
point of the cubic is a finite point, Maclaurin’s method will furnish a real 
means of generating it.] 
§ 15. Prop. IX. 
If, when PQ passes through 0 2 , Q0 2 at the same time coincides with 
0 2 P, the locus degenerates into this line and a conic. 
Cor. 1. This furnishes a means of describing a conic when it is to pass 
through one only of the two points O v 0 2 . 
| 16. Prop. X. 
If 0 X PQ and Q0 2 R are as before, but P and Q are restricted to lie 
on l and V respectively , the locus of R is a cubic possessing a double 
point at 0 V 
