VOL. XLIII.] PHILOSOPHICAL TRANSACTIONS. 21 



motion ceases. In case 2, the moving force of the body is consumed, and its 

 motion ceases before the spring is bent through its whole length, or wholly 

 closed. In case 3, the moving force of the body is consumed, and its motion 

 ceases at the instant that the spring is bent through its whole length, and is en- 

 tirely closed. 



For this reason, and in order to make the following corollaries of more ready 

 use. Dr. J. distributes them into 3 classes; the first of which are as general as 

 the theorem itself, extending to all the 3 cases, but are more particularly useful 

 in case 1. The 2d class of corollaries extends to both the 2d and 3d case; but 

 are more particularly useful in case 2. The 3d class extends only to case 3, and 

 by that means are much more simple than either of the former. 

 Class 1. — General corollaries, but of more particular use in case 1 ; wherein the 

 spring is wholly closed, before the motion of the body ceases. 



Carol. 1. — When the spring is bent through any right sine cb, fig. 7, the loss 

 of velocity is to the original velocity, as the versed sine to the radius, or v — 



« = V X — • For, since i; = vX— , v — i;=v — vX — = vX = 



V. R R R 



Corol. 2. — When the spring is bent through any right sine cb, the diminu- 

 tion of the square of the velocity is to the square of the original velocity, as the 

 square of that right sine to the square of the radius, or v* — d' = v** X 



-^. For, smce v = v X — , w' = v^ X -o, and v' — t;^ = v* — v^ X — r = 



■a.^ R R R 



2 V. R"— BF* 2 v^ CB» 



Corol. 3. — ^When the spring is bent through any space /, v the velocity of the 



body is equal to v X ■/ — ^ , or to v X ^ — | — — ; and is proportional 



- Smlo — pZ' , . 2Ma — pi 

 to V , or to \/ —. 



ML ' M 



For, since y^ = v^ x — r = v^ X — r- ; if for ^ we substitute its value 



R* R' 



2MLa , o 2 ^^ 2MLa— p/» ^^ ,2MLa— pZ* , , _, 



— , we have d^ = v" X — r , or t; = v X v — : and as, by Dr. 



Hook's principle, l : / :: p :/;, or p/ = y>L, t; = v X '^ — j-~^y or i; = v 

 j^ ^ MO — p ^ But -y, by Galileo's doctrine, is a constant quantity; andthere- 



r ■ .^- 1 i ,2MLa— pZ* . ,2ua — pl ' ' 



fore V IS proportional to v , or to v ■-. 



*^ ^ ML M 



Corol. 4. — The time t of bending the spring through any space /, is propor- 

 tional to the arch gf divided by \/a; / being the right sine of the arch, and 



R, = >/ , being the radius. For, by the theorem, ^ = t x ■^•, and — is 



a constant quantity. 



