96 Proceedings of the Royal Society of Edinburgh. [Sess. 
A cubic possessing a double point is a unicursal or rational curve, whose 
freedom equations may be written in the form 
x = A(t)/G(t) ) 
y = B(t)/C(t)f • • • ' ' W 
where A, B, C are integral cubic functions of t. 
They may also be considered as generated by the point common to the 
two straight lines 
Lo + ^i = 0 { (0) 
M 0 + ^M 1 + ^ 2 M 2 = 0 f W 
where L : . . . M 2 are linear functions of x and y. (Vide, e.g., Tweedie, 
“ Courbes Unicursales,” L’ Enseignement Mathematique , 1912.) 
The equation 
L 0 + ifLj = 0 
represents a pencil of lines. 
The equation 
M 0 + tU l + mM) 
represents a system of straight lines whose envelope is the conic 
Mp - 4M 0 M 2 = 0 (3) 
and the corresponding rays of the pencil and the tangents to the conic are 
in projective correspondence. 
§ 9. In this and the next section Maclaurin makes frequent use of a 
constant angle OPQ, where 0 is a fixed point while P is any point on a 
line l. 
In such a case PQ envelops a parabola. For let OP in any position be 
given by the equation 
y-tx = 0 (1) 
The co-ordinates of P on l are then of the form 
fat + b pt + q\ 
\ct + d ’ ct + d) 
The gradient of PQ is also rational and linear in t, so that PQ has an 
equation of the form 
L 0 + 2^L 1 + ^ 2 L 2 = O (2) 
whose envelope is the conic 
L 1 2 -L 0 L 2 = 0 (3) 
in this case a parabola, since, when P is at infinity on l, PQ lies entirely 
at infinity. 
§ 10. Prop. V. 
0 X and 0 2 are two fixed points. The vertex P of the constant angle 
0 1 PQ( = a) lies on a given line l. Q0 2 R is an angle of constant 
