o[) 



2d PHILOSOPHICAL TRANSACTIONS. ( ANNO 1744. 



This is proved in the same manner as corol. 15, putting only l for /. 



CoroL 30. But a quantity of motion less or greater than mv, in any given 

 ratio, may close the same spring, and be wholly consumed in closing it : and the 

 time spent in closing the spring will be respectively less or greater, in the same 

 given ratio. This is easily proved from corol. l6. 



Corol. 31. If the motion of the body cease, when the spring is wholly closed, 



the initial vis viva, or mv', is equal to — — : and 2aM = pl. 



For, by corol. 11, v = c y/- — , or v^ = - — , or mv'' = — — = . 



Carol. 31. In the same case, the initial vis viva is as the rectangle under the 

 strength and length of the spring. 



For, by the preceding corol. mv' =: — — , and — is a given quantity ; therefore 

 MV* is as PL. 



Corol. 33. In the same case, if- be given, the initial vis viva is as p', or as l'. 

 This is evident from the preceding corollary. 



Corol. 34. If the vis viva mv' bend a spring through its whole length, and be 

 consumed in closing it, any other vis viva equal to the former, as tjkm X - will 



nn 



close the same spring, and be consumed thereby. This is evident from corol. 32. 

 Corol. 35. But the time of closing the spring by the vis viva tijim X —, will 



be to the time of closing it by the vis viva mv*, as n to 1. 



For, by corol. 1 1, since the spring is given, the time is as \/m. 



Corol. 36. If the vis viva mv* be wholly consumed in closing a spring of the 

 strength p, and length l ; the vis viva wkmv* will be sufficient to close, 



1. Either a spring of the strength nwp, and length l. 



2. Or a spring of the strength nv, and length wl. 



3. Or of the strength p, and length nwL. 



4. Or if n be a whole number, the number nn of springs, each of the strength 

 V, and length l, one after another. 



For Mv' : «nMv* :: pl : wtjpl ; and therefore, by corol. 32, the vis viva tjkmv' 

 will close any spring that has tzwpl for the product of its strength and length. But 

 nnPL is composed either of nnv X l, or of «p X wl, or of p x wwl. 



Also the loss of the vis viva, in bending a given spring, being always the same, 

 by corol. 7 ; and the vis viva mv'^ being wholly lost in bending a single spring pl ; 

 the loss of the vis viva, nnuv*, in closing one such spring, will be mv'^; and its 

 loss in closing a second such spring, will again be mv'^, and so on : consequently 

 the number nn of such springs will be closed one after another, by the time that 

 the vis viva nnMv"^ is wholly consumed. 



Scholium. — If the spring, instead of being at first wholly unbent, as we have 



