The following results were obtained lately while I was considering 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 coplanar 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 intersection of AP and BQ.