the Times of Vibration of Watch Balances. 147 
daily ra+e of the time-keeper may be affected by this altera- 
tion in the position of the quiescent points, remains unknown, 
unless it be investigated by assuming the deviation of the points 
of quiescence, as one of the conditions on which the time of vi- 
bration depends. This condition is included in the preceding 
investigations, which may be now applied to the solution of 
some cases, which are suggested by considering the construc- 
tion of Mr. Mudge’s time-keeper. 
Any minute or particular description of this ingenious in- 
vention would be foreign to the subject of theoretic investiga- 
tion ; an outline only of the construction on which the action 
of the several springs depends will be necessary, to render the 
application of the preceding theorems sufficiently intelligible. 
ONEBQ (Tab. XV. fig. 8. ) is the circumference of the ba- 
lance, vibrating by the action of a spiral spring on an axis C A 
D H, passing through the centre C ; the axis is discontinued 
from A to D to make room for the other parts of the work. C A 
and D H are connected by means of a branch or crank A X 
Y D, which is fixed to the axis C A D H, and always vibrates 
with the balance on the said axis.* 
L M, Z W, are two rods affixed to the crank at the points L 
and Z parallel to X Y ; which rods also vibrate with the balance. 
c, d, e,f g, r, s, are fixed parts of the machine. T R is an axis 
in the same right line with C A D H carrying an arm G O at 
right angles to itf (or nearly so), and a small auxiliary spring 
u, which is wound up whenever the arm G O is turned round 
* The additional weight affixed to the balance at t counterpoises the weight of the 
crank or branch A X Y D, so as to bring the centre of gravity of the whole into the 
axis of motion. 
t In the machine G O and I O are not exactly at right angles to T R and F S ; but 
are so represented in the figure, in order to make the different positions of the arms 
G O and I O the more distinct. 
U 2 
