140 Mr. Atwood's Investigations for determining 
TTP ; and the velocity of the circumference at R (fig. 4-) = 
/ 2 If 
X </b z -f- nc z — x z : let t be the time of describing 
QR ; then i = 
and t = -4y x into 
■ x 1 2 l J 
2 If * v/ b* + n 
a circular arc of which the cosine is ===== to radius = 1, 
\/ b -f- n c 
which should = o when x = d; wherefore t or the time of 
describing QR = * into a circular arc of which the 
cosine is 
a x into a circular arc of which the 
2lf 
\/b* + n c 2 
cosine is d — - ; and when x = 0 , that is when the entire 
■ x into a cir- 
s/b x + n c 
arc Q O has been described, the time t 
cular arc of which the sine is ^====- 
The result of this investigation is, that the time in which 
the balance describes the semiarc B O, by the joint action of 
both springs through the arc B Q, and by the action of the 
balance spring only through the arc O O 
is = \/ 
2 a — x an arc, of which the sine is \/ V 
//xn+i 2 b + 2,lC 
+ 
- j x an arc, of which the sine is y ^ + ~ 
(to radius = 1 ) expressed in parts of a second.* 
* When n - o, this expression becomes y/ZIl X "twice an arc, of which the siiii 
is v/Z + an arc, of which the sine is -f : but since c = b _ d, the two arcs tere men, 
tinned’ will be exactly = 90° = f ; and the time of a semivibration = X T" 
- v/^E, agreeing with the solution in page 126. Suppose d - o ; since in this 
case b — c, the time of a semivibration becomes ^ jp-r== X an arc, of which the 
sine is s/Zfl which arc is = 45 0 ,orf 1 wherefore the time of a semivibration in 
this case = ^ — * p% ■ which is. the true value, according to the solution in page 
8//X» + i 
126. See also page 146. 
