1 6 ^ Mr. Atwood’s Investigations for determining 
on the position of the balance or crank A X Y D ; if the rod 
LM should be removed (by turning the balance in its plane), 
suppose that the arm GO rests in a position GN, at a distance 
beyond O, which is measured by the arc ON = i°; in this 
position the point of quiescence of the auxiliary spring u will 
be situated in the latter semiarc of vibration at N, between E 
and O, and by a similar adjustment of the quiescent position of 
the arm IO, (fig. 5 and 8.) the point of quiescence of the auxi- 
liary spring v will be situated at Q in the latter semiarc of the 
vibration, between O and B ; in consequence of this position 
of the points of quiescence, the daily rate of the watch will be 
accelerated i9".38, (p. 159) while the semiarc of vibration 
continues 135 0 , and when the semiarc of vibration is dimi- 
nished to 125 0 , the daily rate will be further accelerated ; the 
rate being 20". 91 fast, (page 160) or i"53 faster than when 
the semiarc of vibration was 135°. 
From these several* results it may be concluded, that al- 
though the rate of going of Mr. Mudge’s time-keeper depends 
materially on the quiescent position of the arms GO, I O, 
(fig. 8.), that is, on the position of the points of quiescence of 
the auxiliary springs, yet while that position remains unaltered, 
whatever it may be, the regularity of the time-keeper will not 
be affected in consequence of the said position, provided the 
semiarc of vibration continues the same ; but when this arc is 
*- It is scarcely necessary to observe, that although these results have been investi- 
gated from supposing the elastic forces of the spiral springs to be in the precise law of 
the tensions, or distances from quiescence T yet if the spring’s forces should deviate 
somewhat from that law, the general conclusions deduced from the preceding calcu- 
lations, respecting the acceleration or retardation of the rate arising from an alteration 
in the position of the points of quiescence, will still be true, although the degree in 
which these effects take place may not be exactly the same as when the spring’s forces, 
see. in the precise law assumed in the investigations. 
