126 Mr. Atwood’s Investigations for determining 
and taking the fluents while x decreases from b to x, 
u — F * b L~ _ ; if therefore l is made = 193 inches, being 
the space which bodies falling freely from rest by the force 
of gravity near the earth’s surface describe in one second of 
time, the velocity of the circumference, when the extremity A 
of the index C A has arrived at the point H, will be 
■ = x s/v—x\ 
Let t represent the time in which the circumference describes 
the arc B H ; then will t = \/ - jp x ^== » and 1 ~ 
x into a circular arc, of which the cosine = | to 
radius == 1, which is the time of describing the arc B H ex- 
pressed in parts of a second ; when x — o, that is when the 
circumference has described the entire semiarc B O, the circu- 
lar arc of which the cosine = y is a quadrant of a circle to 
radius =1. Let p = 3.14159* &c - The time t, of describing 
^ ^ ^ p p Z U 
the semiarc BO = \/ x — = \/ 8 1 
In this expression for the time of a semivibration, the letter 
« denotes the length of the arc OD (fig. 1.) ; if this arc should 
be expressed by a number of degrees c , a will then = ~ jg 0 ° » 
and this quantity being substituted for a , the time of a semi- 
vibration will be t ; if instea d of F, its v alue 
is substituted in the equation t = the 
time of a semivibration will be t = - g ^ ^ 
Let the given arc c° be = 90° ; in this case t — lb ? f~' 
expression the lines w and * are supposed to increase together ; but if u increases while 
x decreases, the signs of the variations » and x will be contrary ; in which case the 
equation will become u ~ — Hx. 
