20 PHILOSOPHICAL TRANSACTIONS. [aNNO 1744: 



time of a heavy body's ascending in vacuo with the velocity v, as the corres- 

 ponding arch to 2a : that is < = t X — • 



Demonstr. — 1. While the spring is bending through the space cb, let the 

 space, through which it is at any time bent, be called x, r the time of bending 

 it through the space x, and u the velocity of the body at the time t ; and let cl 

 := L, CB = /. Then, if p be the force, with which the spring, when bent 



through the space x, resists the motion of the body ; by Dr. Hook's principle, 



pj 

 l: a; :: p : ft^ — , 

 "^ L 



And since, in the case of forces that act uniformly, the quantities of motion 

 generated are proportional to the generating forces, and the times jointly, if mO 

 be the nascent quantity of motion taken from the body by the resistance 



— m the nascent time t, mv : — Mu :: mt :: — , or — u ^ 



I. ' MLT 



Also, since, in the same case of forces acting uniformly, the spaces are pro- 



2w 



portional to the velocities, and the times jointly, x : 2a :: ut : vt, or t := — -^. 



Therefore, — u = ^^ X — , or 2uu = — 1^^; and the fluents of these 



' MLT 2au JILO 



two quantities are u'^ and — x— - • But the former of these was v% when x, and 



T 2MLa 



consequently the latter, was nothing; therefore. 



VPX* , , v'pr 



w^ - v^ = - -^, or .^ = v^ - , . 



Smlo 2m La 



-n . 1 • 2m Lo .1 1 /• <) o v'x* o a 



But, by the construction, = »■*; therefore, v = \-' j-, or u = v" 



X ^ ~' ; and, when x becomes equal to /, and u to v, u^ = v^ X ,» ■ ; and, 

 by the property of the circle, r'^ — P being equal to 

 bf', v'' = v" X ^, or i; = V X — . a. k. d. 1°. 



' R^ K 



2. We have above, t = — ; and u-" = v* X — ^t" » or u = v X : 



therefore r = — X TT:;?^?-^^. ">• ^ = 2a >< V^TI^'- 



Now let CD, fig. 8, be equal to x; and draw the co-sine de, the radius ce, 

 and the perpendicular ed = i; then will the triangle dec be similar to the nas- 

 cent triangle deE ; and consequently he : de :: ce : eE = 

 ^Jl2SJ1= - y— — - . Therefore, t = ~ X ce, and t= t X ^. And when 



DE ^^R* — x» ' 2o ' 2a 



a becomes equal to cb, and r to t, the arch ge becomes equal to the arch gp: 

 therefore i = t x |^. a. e. d. 2°. 



Under this theorem are comprehended the 3 following cases: 

 In case 1, the spring is bent through its whole length, or is entirely com- 

 pressed and closed, before the moving foice of the body is consumed, and its 



