108 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Therefore S generates a straight line V , and BS, parallel to 0 2 T, is 
perpendicular to SQ. 
But B is fixed, S lies on l ' , BSQ = 0 2 QS = .*. etc. 
[Since PN in the original construction is always tangent to a parabola, 
the constant angle /3 shows that the locus of Q is simply the oblique pedal 
of a parabola and falls to be discussed in Part II as a pedal.] 
We now assume 
o 
a = P = y 
Maclaurin notes when 0 2 is a node, a conjugate point, or a cusp. When l 
passes through 0 2 and is ± r 0 1 0 2 the curve is the cissoid. 
§19. Equation to the Curve. 
Choose the origin at 0 2 . 
Let O-l be the point (a, b ) ; and let the equation to l be 
Let P be the point 
so that the gradient of O x P is 
and PQ has the equation 
while 0 2 Q has the equation 
y = mx + n 
(6 m$ + n), 
+ n-b 
y-m£-n= “ ^ - (x - £) 
m$+n - o 
mP -bn - b 
y — — x 
£ - a 
(i) 
( 2 ) 
(3) 
To obtain the locus of Q, eliminate £ between (2) and (3). 
. (y — mx)(x 2 + y 2 ) + x 2 (b — n) + xy(bm — a) - y 2 (am + n) = 0 . . (4) 
Now any circular cubic with double point at 0 may be written as 
(y - Xx)(x 2 + y 2 ) + Ax 2 + Bxy + Cy 2 = 0 . . . . (5) 
If (4) and (5) represent the same curve, we must have 
m = \ ' 
h b ~ n=A v ^ 
bm - a= B 
— am -n=C 
These determine m, n, a, b uniquely, corresponding to any equation (5). 
Maclaurin’s generation therefore furnishes all the circular cubics. 
To the lemma Maclaurin attaches several corollaries of special interest. 
Cor. 1. Cissoidal Generation of the Curve. 
