28 PHILOSOPHICAL TRANSACTIONS. [aNNO 1744. 



the space bd = i, that is, if £ = O; then the height to which the body might 

 ascend in vacuo, with the velocity v, or ut = . 



For, by the last, when i = 0, 2aML = p* X 2/ + *. 



Corol. 39. Up, the force of the spring when bent through the space cb, be 

 equal to m the weight of the body ; the height to which the body would ascend 

 in vacuo with the velocity v, is to the space through which it will bend the 

 spring, by striking it at b with that same velocity, as 2/ -j- * to 2/, or mis :: 2/ 

 4- * : 2/. 



For, by the last, a = — i ; and - being equal to y, « =: ^-^ — j-— ; and 



•r ^ v> 2/+ « 



\tp = M, a.= S X — g— . 



Corol. 40. If /) = M, and p also continue constantly the same, while the 

 spring is bending from b to d (both which suppositions are necessarily made in 

 reducing the action of a spring to that of gravity on an ascending body), the 

 spring must be of an infinite length ; and /, the space through which it was bent 

 before the body struck it, must also be of an infinite length ; and the space bd, 

 through which the spring will be further bent, must be equal to the height the 

 body can ascend to with the velocity v, or x = s. 



For, by the last, when p = m, a. : s :: 2l -\- s : 2l ; and the resistances of the 

 spring at d and b being respectively as cd and cb, that is, as / -j- 5 and / ; since 

 those resistances are now supposed equal to one another, we must, on that sup- 

 position, consider / + * as equal to / ; and adding / to each, 2/ If * = 2/, that is, 

 / must be infinitely greater than s ; and then a : 5 :: 2/ : 2/, or a = j. 



In this proposition, and all its corollaries, except the last 4, we have considered 

 the spring as being at first wholly unbent, and then acted on by a body moving 

 with the velocity v, which bends it through some certain space : but as we sup 

 pose the spring to be perfectly elastic, the proposition and corollaries will equally 

 hold, if the spring be supposed to have been at first bent through that same 

 space, and, by unbending itself, to press on a body at rest, and thereby to drive 

 that body before it during the time of its expansion : only, v, instead of being 

 the initial velocity with which the body struck the spring, will now be the final 

 velocity with which the body parts from the spring, when wholly expanded. 



If the spring, instead of being pressed inwards, be drawn outwards by the 

 action of the body, we need only make l the greatest length to which the spring 

 can be drawn out beyond its natural situation, without prejudice to its elasticity, 

 /any less length to which the spring is drawn outwards, p and/) the forces which 

 will keep it fi-om restoring itself, when drawn out to those lengths respectively, 

 and the proposition will equally hold good : as it will also, if the spring be sup 



