1915-16.] The “ Geometria Organica ” of Colin Maclaurin. Ill 
x{x 2 + y 2 ) - 2 a(x 2 + y 2 ) + (2aa - a 2 — /3 2 )# + 2a/3y = 0. 
(0 2 is the point (a, /3) and l is the line x — a — 0.) 
Teixeira points out that the identical locus is discussed by Lagrange 
(. Nouvelles Annates, 1900), and that the equation represents part of the 
curves known as Van Rees’ Focals, for which the equation may be 
reduced to 
x(x 2 + y 2 ) = A (a? 2 + y 2 ) + Rr + Gy. 
Maclaurin shows that the curve has a closed oval and a serpentine 
branch save (Cor. 8) when 0 2 is on the line l, when there is a node. 
Cor. 10. If 0 2 is on l, and D0 2 perpendicular to l, the curve is that 
described by De Moivre in No. 345 of the Philosophical Transactions. 
Cor. 11. The strophoid may also be thus generated : — 
D and 0 2 are fixed points, and 0 2 P a fixed line l. 
PQD is a constant angle = P0 2 D, and PQ = D0 2 . Then, as P slides on l, 
point Q generates the strophoid (R0 2 = RQ). 
Also the mid-point of PQ generates the cissoid of Diodes when 
PQD = ^ (Newton). 
[The description of the strophoid as the intersection of two rays rotat- 
ing round two fixed centres with angular velocities in the ratio 1 : 2 is 
ascribed to Plateau (1828) by Kohn and Loria in their article on Special 
Plane Curves in the Encyk. der. Math. Wiss.- This is historically inaccurate, 
for Maclaurin gave this generation three-quarters of a century earlier in 
his Fluxions (p. 262).] 
§ 20. Prop. XII 
discusses the asymptotes and also the subvarieties of the curves of Prop. X. 
Ex. 1. Let a = /3= IP O^; V II 1 0 1 0 2 . 
If O x is the origin, 0 2 the point (d, 0) ; equation to l, x — a\ equation 
to V, y = b ; the locus of R is given by 
ay 2 (x - d) + bdxy = (d- a)x 2 (x - d). 
The case l and V both parallel to 0 1 0 2 is discussed in Ex. 4. 
