436 
Proceedings of the lloycil Society 
of the former equation being, by the help of the latter, at once 
obtained in the form 
Tp -f T(<n — p) = constant. 
2. On the Ovals of Descartes. 
The following results were obtained lately while I was consider- 
iug how most simply to describe by working sections surfaces 
analogous to that treated in the preceding note. They are so 
elementary that it is not likely that they can be new, but as they 
are novel to myself, and to several mathematicians whom I have 
consulted, I bring them before the Society : — 
Let two coplan ar circles be described, with centres A and B. 
Take any point, C, in the line of centres, and draw a line CPQ, 
cutting the circles in P and Q. Find the locus of R, the inter- 
section of AP and BQ. 
Expressing that CPQ is a straight line, we have, if 0 and </> be 
the angles at A and B respectively, 
AP sin 0 ^ BQ sin <j> 
AP cos 0 =b AC BC rb BQ cos <f> 
or 
AP . BC sin 0 rb AO . BQ sin <f> = rb AP • BQ sin (0 + <£) , 
which, by substituting the sides of ABB for the sines of the angles 
opposite them, becomes 
AP . BC • BR rb AC • BQ • AR = rb AP . BQ. AB (1) 
which is the general equation of Cartesian Ovals. 
