362 Proceedings of the Royal Society 
One particular case of this motion, viz., the case of the swing- 
ing of a clock-pendulum, is of paramount importance, and has been 
investigated with very great care. In this case our attention is 
directed principally to the computation of the time of an entire 
oscillation, since it is this which determines the heating of the 
clock. In the paper to which this note is an addition (Yol. xxiv. 
Trans.), a very rapid method of computing this total time is 
given. My object is now to supply the deficiency in that paper, 
and to show how the time of describing any given portion of the 
whole arc may be computed. 
The general question may be stated thus: — A heavy body is 
projected with a known velocity along the circumference of a circle, 
and we are required to compute the time in which it will reach 
any indicated position, as also its place at any prescribed time. 
No practicable solution of either of these problems has hitherto 
been given, with the exception of the case already mentioned. 
This note contains a simple and complete solution of both 
problems. 
If a heavy body be projected from the lowest point of a circle 
along the circumference with a velocity less than that due to a fall 
from the highest point, its motion becomes slower as it ascends, 
and its speed is entirely exhausted at some point in the semi- 
circumference ; from that point it returns to the bottom of the 
curve, passes to the other side, and so oscillates. But, if the 
initial velocity he greater than what is due to a fall along the 
diameter, the body passes the zenith point, and circulates round 
and round the circumference with an unequable motion. And if the 
velocity be just sufficient to carry the body to the zenith point, it 
rests there, and the motion ceases. Now, while the investigation 
of the oscillatory and of the continuous motion is difficult, that of 
the limit between .the two is easy. 
If the body move away from N with a velocity due to a fall 
through the distance ZN, it will have, when it reaches the point 
A, a velocity due to a fall through ZGr, But the distance through 
which a weight falls freely is proportional to the square of its 
acquired velocity, and ZGr is proportional to the square of ZA ; 
wherefore the velocity at the point A must be proportional to the 
chord ZA ; that is to say, the rate of increase of the angle NZA is 
