04 PHISOSOPHICAL TRANSACTIONS. [aNNO 1744. 



Carol. 16. But a quantity of motion less than mv, in any given ratio, may 

 bend the same spring through the same space /, and the time of bending it will 

 be less in the same given ratio. 



For, let 1 to n be the given ratio ; and let the less quantity of motion be 



— X nv ; which is to mv, as 1 to n. Then by corol. 14, the spring being given, 



,M M . MV . . I / M MV 



/\/m: /y/— :: mv : - X wv = —■xl'/—= — • Therefore the quantity of 



motion — x n\, being equal to — , will bend the spring through the same 

 space /. 



Also, by the same corollary, mv is as /i ; and / being given, the quantity of 

 motion is as < : therefore the time of bending the spring will be less in the same 

 ratio, as the quantity of motion is less. 



Corol. 17. A quantity of motion greater than mv, in any ratio given, may 

 be consumed in bending the spring through the same space ; and the time of 

 bending it will be greater in the same given ratio. 



This appears after the same manner as the preceding, by making n a fractional 

 number, instead of a whole one. 



Corol. 18. If the motion of the body cease, when the spring is bent through 

 any space /, the initial vis viva, or mv^, is equal to 



— i— , or to — : and 2 qm = — = ft/. 

 2la ' 2a l ^ 



For, by corol. 8, v = cW^r^— = c/-^, or v* = ^— = f^' : There 



fore MV-' =. — = -£- = 

 2la 2a 



Corol. 19. In the same case, the initial vis viva is proportional to— = pi; 

 and if -be given, the vis viva is as P, or as jB'. 



For in the preceding corol. — being a given quantity, the vis viva is as — := 



pi ; and if- be given, it will be as /'■', or as p"^; as p and / increase in the same 



proportion. 



Carol. 20. If the vis viva, mv% bend a spring through the space /, and be 

 totally consumed thereby ; any other vis viva, equal to the former, as miM x 

 — , will bend the same spring through the same space, and be totally consumed 

 thereby. 



For, the spring being the same,- is given ; and therefore by corol. 19, the 

 vis viva which will be consumed in bending the spring through the space /, is as P. 



Corol. 21. But the time in which the same spring will be bent through the 



