408 
Proceedings of the Royal Society 
4 . On Transversals. By the Kev. Hugh Martin, Free 
Greyfriars. 
This paper contains upwards of seventy theorems, in great part 
new, with reference to the intersection of systems of lines. The 
demonstrations are based upon determinants. 
5 On the Motion of a Heavy Body along the circumference 
of a Circle. By Edward Sang. 
This paper contains a demonstration'of the theorem given in the 
fourth volume of the proceedings at p. 419. 
The theorem in question was arrived at by the comparison of the 
well-known formula for the time of descent in a circular arc, with 
another formula given in the “ Edinburgh Philosophical Magazine ” 
for November 1828, by a writer under the signature T. W. L. 
Each of these series is reached by a long train of transformations, 
developments, and integrations, which require great familiarity 
with the most advanced branches of the higher calculus. Yet the 
theorem which results from their comparison has an aspect of 
extreme simplicity, and seems as if it could be reached by an easier 
road. A search for this easier road has led to the discovery of a 
few simple propositions, which contain the whole theory of motion 
in a circle, and which only require for their examination a know- 
ledge of the fundamental law of dynamics, and an acquaintance 
with trigonometry. 
There are two distinct cases of motion in a circle, viz., one in 
which the heavy body describes with a varying velocity the whole 
circumference ; the other in which the motion is oscillatory, as in 
the example of a pendulum. The first step of the investigation is 
to establish a relation between a continuous and an oscillatory 
motion, such that their periodic times bear a certain ratio to each 
other. To these two motions the name conjugate is given. By 
means of this relation the problem, “ to compute the time of re- 
volution” in the case of continuous motion, can be converted into 
another problem, “ to find the descent in a circular arc,” and 
contrariwise. 
