674] 
26] 
674. 
NOTE ON THE CONSTRUCTION OF CARTESIANS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878), p. 34.] 
If p = a + h cos 6, and r = ^ {p + f(p 2 — c 2 )}, then obviously r 2 — rp + ¿c 2 = 0, that is, 
r 2 —r(a+b cos 6) + ¿c 2 = 0, 
which is the equation of a Cartesian. Here p = a + b cos 6 is the equation of a 
limaçon or nodal Cartesian, having the origin for the node ; and for any given value 
of 6, deducing from the radius vector of the limaçon the new radius vector r by 
the above formula r = \ {p + f(p 2 — c 2 )}, we obtain a Cartesian, or by giving different 
values to c, a series of Cartesians having the origin for a common focus. The con 
struction is a very convenient one.