609 
of Edinburgh , Session 1871-72. 
to pass from the second to the third as it takes to pass from the 
first to the second. It suggested to me the following theorem, 
which really involves Mr Sang’s results, hut which appears to be 
considerably simpler in treatment, this being my sole reason 
for bringing it before the Society. 
Let DM be a horizontal line, and let DA be taken equal to the 
tangent from D to the circle BPC', whose centre C is vertically under 
D. Also let PAQ be any line through A, cutting in Q the semi- 
circle on AO. Also make E the image of A in DM. Then if P 
move with velocity due to DM, Q moves with velocity due to the 
level of E ; so that we have the means of comparing, arc for arc, 
two different continuous forms of pendulum motion, in one of which 
the rotation takes place in half the time of that in the other. 
Let to be a small increment of the circular measure of BAP, then 
arc at Q = CA . co , arc at P = 
AP. PC 
PQ 
CO . 
Also, 
velocity at P = J 2g . PM = ,J • AP . 
Hence, 
velocity at Q = ^PQj^ >Ap 
9 - AC 
PC 
• PQ. 
But 
PQ = VOP 2 - CQ 3 
= »/CP 2 — CR . CA (where QR is horizontal) 
, /ftps _ PA a 
= JCAj - - CA + AR = JCA . ER . 
Hence, 
AO 
velocity at Q = ^Q-Jg . ER. 
Thus Q moves with velocity due to the level of E, and constant 
acceleration 
AC 2 
2P02 -3- 
The second process referred to above gives at once the means 
of comparing continuous rotation with oscillation, as follows — 
