330 Mr. Atwood's Investigations for determining 
force, instead of varying in the ratio of the aforesaid distances, 
should vary in the T 9 ^ power, or ^ power of the distances, 
it does not appear from the preceding solution what alteration 
in the daily rate would be caused by this change in the law of the 
force's variation, when the semiarc of vibration is increased or 
diminished by a- given arc. To ascertain this point fully, other 
researches will be necessary, by which it may be known, what 
alteration of the daily rate of a time-keeper is occasioned by a 
given increase or diminution of the arc of vibration, when the 
spring's elastic force varies in a ratio of the distances from the 
quiescent position, the general index or exponent of which is 
any number or fraction n. 
The force which accelerates the balance being assumed in 
that power of the distances the exponent of which is n, let 
g O == b (fig. 3.) l>e the semiarc of vibration when the time- 
keeper is adjusted to mean time ; let D O =a ; the accelerat- 
ing force on the circumference at the distance from quiescence 
OD=F; suppose the circumference to have described the 
arc B H from the extremity of the arc B ; and let HO = i; 
then the force by which the circumference is accelerated 
when at the angular distance from the quiescent position 
q pj _ F - ga ; let u be the space through which a body falls 
freely from rest by the acceleration of gravity, to acquire the 
velocity of the circumference when it has described the arc 
B H ; the principles of acceleration give this equation : 
^ ; taking the fluents while x decreases from b to .z, 
U = L^tlz F - r — , and l being 193 inches, the velocity ac- 
re -J- l X a n 
quired by the circumference after describing B H, will be 
