the Times of Vibration of Watch Balances. 
131 
=v ^ZIx/F 
+ 1 • 
n + 1 x a n 
let T be the time of describing the arc B H ; wherefore 
T=V- 
+ 1 X a n 
4^ 
s/ b n + ‘ 
€" + 1 
The time of describing the arc B H will be the fluent of this 
fluxion, while x decreases from b to x, and the time of de- 
scribing the semiarc B O will be the entire fluent of 
v/I 
+ 1 X a” 
4 1 F s/b” + l -x n + 
Now let the balance commence its vibration from any other 
point I, (fig. 3.) and let IO = c; suppose the circumference to 
have described the arc I K, and make OK =y ; let t be the 
time of describing the arc IK; then by proceeding in the same 
manner as in the former case, it is found that i = . n + 1 * an 
4 /f 
=, while x decreases from b to 0 . 
* v /p = p=3^qn ; and the time of describing the semiarc I O, 
will be the entire fluent of this fluxion, while y decreases from 
c to 0. Although the fluents of the fluxions ^= ^7, ==^= = and 
— y 
y/ V- 4- ■ _ y tzp cannot be expressed in general terms, yet the 
exact proportion of the said fluents may be assigned, which 
will be the proportion of the times in which the balance vi- 
brates in the two semiarcs BO, IO; the multiplying quantity 
v/ ” being common to both fluxions ; and since the 
4/F 
entire fluent* of , ==== = = === == 
s /b n + l — x n + x 
is to the entire fluent of 
