VOL. XLIII.] PHILOSOPHICAL TRANSACTIONS. Q.'J 



hitherto considered it, be now supposed to have been already bent through some 

 space CB, before the body strikes it ; and the velocity of the body be required 

 after the spring is bent through any further space, bd, fig. 1 1, this case, as well 

 as the three others above mentioned, will be found to come under our theorem. 



For if f be the velocity with which the body is supposed to strike against the 

 bent spring at b ; it is evident that this may be considered, either as the original 

 velocity, or as the remainder of a greater velocity v, with which the body might 

 have struck on the spring at c, and which, on bending the spring from c to b, 

 would now be reduced to v. For in either case the effect in bending the spring 

 from B to D, will be exactly the same. 



In order therefore to determine this imaginary velocity v, let a middle propor- 

 tional, BF, be taken between cl X -, and 1x, « being the height to which a 

 body will ascend in vacuo with the velocity v ; draw bf perpendicular to cb, and 

 with the radius cf describe the quadrant cgfea. Then will our present case be 

 exactly reduced to that of the theorem ; cb and cd representing the spaces 

 through which the spring is bent ; bp and de the velocities in the points b and 

 D ; GF and ge the times of bending the spring through the spaces cb, cd ; and 

 CG representing the imaginary velocity v, with which the body might have struck 

 the spring at c. 



For, by the theorem, bf* : cg' :: v^w"^; and u* : v- :: a : a. Therefore cg' = bf* 

 X -. But BF^ = 2a X — , by the construction ; and consequently CG^ ^ — — 



X - = , as in the construction of the theorem. 



» r 



From this case we shall draw a few corollaries, as well for their usefulness 

 on other occasions, as to show how the theory of springs may be safely applied 

 to the action of gravity on ascending or falling bodies. 



Carol. 37. If the body m, with the velocity v, sufficient to carry it to the 

 height a, strike at b, on a string already bent through the space cb = /; and do 

 thereby bend it through some further space bd = j ; at the end of which space, 

 or at D, the body has a velocity sufficient to carry it to some height, as f ; then 



2«ML — pf X 2/ + S 



2ml ■ 



For, by the theorem, 



« : I :: bp'' : de*, or de* = bf" X - = X -, or de* = 



p « p 



Also, DE* + CD* = CE- = CP* = BP* + CB*, that is, ?!^ -f /* + 2/i 4- i* = 

 2«MI, , „ 2»ML 2«ML „, Q „ „ -r—. — 



— - + r; or = 2ls — r; or 2fML = 2aML — fs X 2l -{■ s. 



Carol. 38. If the motion of the body cease on bending the spring through 



E 2 



