11 PHILOSOPHICAL TRANSACTIONS. [aNNO 1744. 



Carol. 5.-r-The diminution of the product of the weight of the body into the 

 square of the velocity, or, to use the expression of some late writers, the diminu- 

 tion of the vis viva, that is, mv^ — iav\ by bending a spring through any space 

 I, is always equal to - — , or to -^; where a is the height from which a heavy 

 body will fall in vacuo in a second of time, and c is the celerity gained by 

 that fall. For, by corol. 1, v^ — i;^ = v^ X — j = — j- ; and r^, by the con- 



It R ^ 



struction, being equal to -^-, v'^ — v^ = v^ P X r^. But, by Galileo's theory, 



— = -: therefore v* — i;-' = , and mv* — mv = = -i- . 



a A 2mla 2la 2a 



Corol. 6. — ^The diminution of the vis viva, by bending a spring through any 



p/' 



space /, is always proportional to — , or to pli and if either the spring be given, 

 or - be given in different springs, the loss of the vis viva will be as P, or as p"^. 

 For, by corol. 5, mv^ — mw' = -^^ = —-- ; and ^ being a constant quantity, 

 Mv^ — uv"^ is as — = pi: and if - be given, mv'^ — mi>^ will be as P; 



orasP X-,; or as P X j,; or as p'\ 



Corol. 7. — ^The loss of the vis viva, by bending a spring through its whole 

 length, or by wholly closing it, is equal to — — , and is proportional to pl : and 



if PL be given, the loss of the vis viva is always the same. This is evident from 

 corol. 5 and 6, for / is now equal to l. 



Class 1. — Corollaries of more particular use in Case 1 ; wherein the motion of 

 the body ceases before the spring is wholly closed. 

 Corol. 8. — If the motion of the body cease when the spring is bent through 

 any space /, the initial velocity v is equal to c/ ^/- — , or to cV — . For, by 



corol. 5, v^ — v^ = = r^. And here, the motion of the body ceasing, 



d" = O. Therefore v^ = ^-^ = ^; or v = cW -^— = c/-^. 



2mla 2ma 2mla ' 2ma 



Corol. 10. — If the motion of the body cease when the spring is bent through 

 any space /, the time t, of bending it, is equal to \" of time, multiplied by 

 ^^IlL or to \" X ^\^7r-, where m is to 1, as the circumference of a circle to 



2 * 2pa' 2 2pA 



the diameter. For, by the theorem, 



t = rx^; an^, by Galileo's theory, -1 = ^. Therefore t = -f^ X ^^. 



R* — /' 



Also, by the theorem t;^ = v'^ X — 5 — ; and therefore v^ being now equal 

 to 0, r'* = P, and, fig. 9, / becomes the radius of the circle; and / being also 





