Mr Sang's papers in recent parts of the Transactions of the Society have reminded me of some geometrical constructions which are to a certain extent indicated in Tait and Steele's Dynamics of a Particle (1856). Some of these were suggested to me by a beautiful construction given (I believe by Clerk-Maxwell) in the Cambridge and Dublin Math. Journal, Feb. 1854, the others by a very simple process which occurred to me for the treatment of oscillations in cycloidal arcs. The former enables us easily to divide the arc of oscillation of a pendulum, or the whole circumference if the motion be continuous, into two, four, eight, &c., parts, which are described in equal times; also to solve by simple geometrical constructions problems such as the following: —Given any three points in a circle, find how it must be placed that a heavy particle, starting from rest at one of them, may take twice as long to pass from the second to the third as it takes to pass from the first to the second.