136 Proceedings of the Royal Society of Edinburgh. [Sess. 
Then 
COjOg = R'OgOj - R"0 3 0 1} 
and 
o 1 c=o,R' = o,ir 
Hence 
0 3 R ,/ || 1 0 1 C and = 0 1 C; 
I^OOi and = 00, : 
so that 
OR" 1 1 1 and = 0 1 0 3 = r. 
Also 
CR'IPOA. 
Hence 
OR"R' - CR'R" = tt/2, 
and the locus traced by R' is the pedal of the circle of centre O and radius r 
for a pole at C. 
Cor. 7. The limaqon cannot be rectified ( vide Nicole in Actis Academics 
Parisiensis, 1707). 
Cor. 8. It is a conchoid of the circle whose diameter is OC. For let 
OP cut this circle in G. Hence PG = PO-f OG = r — d cos 6 Ad cos 6 — r, 
and the curve is a conchoid for the pole at 0 and the constant r. 
“ From this it is obvious that the curve is the conchoid of circular base 
described by De la Hire in 1708, and which Roberval and Pascal have also 
discussed. None of these, however, as far as I know, observed that it 
is an Epicycloid (‘ Eorum tamen nemo quantum novi earn Epicycloidem 
esse observavit 
Loria ( Ebene Kurven ) ascribes the discovery of this property to 
Cramer, but clearly Maclaurin has a prior claim. It is one of the few 
occasions on which he does lay claim to a discovery, and he should certainly 
be credited with it. 
Cor. 9. The mid-points of the chords through 0 lie on a circle. 
Generalisation. 
Cor. 10. Take two fixed points O and O', P any point on the circle 
x 1 + y 2 + 2gx + 2/y + c — 0 . . . . ( 1 ) 
Let OPT be a constant angle, which, without loss of generalisation, 
Maclaurin takes to be a right angle. Draw O'T ± r PT. 
If the origin is chosen at O and O' is the point (d, o), the equation to 
the locus of T is 
( x 2 + y 2 y + (2 gx + 2 fy)(oc 2 + y 2 ) 
+ cx 2 + y\d? + c + 2gd) - 2fdxy = 0 .... (2)* 
* The pedals of the central conics give rise to the rational bicircular quartics (Khon 
Loria, Encykl. d. Math. Wiss.). 
