VOL. XLIII.] PHILOSOPHICAL TRANSACTIONS. 25 



same space, by the vis viva nm/i X — , will be to the time in which it is so bent 



' •' nn 



by the vis viva m X v^, as n to 1 ; n being any whole or fractional number. 

 For, by corol. 11, since -is given, the time is as vm. 



Class 3. — Corollaries in Case 3 ; wherein the motion of the Body ceases, at the 

 Instant that the Spring is wholly closed. 

 Corol. 11. If the motion of the body cease, when the spring is bent through 



its whole length, or is wholly closed, the initial velocity v is equal \.o c*/ - — . 



pi 



For, by corol. 8, v = c\/~—\ and / being now equal to l, (fig. 10) j& be- 

 comes equal to p ; and therefore v = c^/- — . 



Corol. 23. In the same case, the initial velocity v is proportional to 

 y'— . For -r-, in the preceding corollary, is a given quantity. 



Corol. 14. In the same case, if pl be given, either in the same, or in dif- 

 ferent springs, the initial velocity v is reciprocally as y'M. This is plain from 

 the preceding corollary. 



Corol. 15. If the motion of the body cease, when the spring is wholly closed, 

 the product of the initial velocity, and the time spent in closing the spring, or 



\t, is equal to 1* X -7 — ; and is proportional to l, the length of the spring. 

 For, by corol. 11, v = c\/- — ; and, by corol. 10, t = I" X -z>y - — : there- 



' ■' ' 2ma ' ■' ' 2 2pa 



fore vi =: 1" X -7--; and \", m, and-, being given quantities, v< is as l. 



Corol. 26. In the same case, the initial quantity of motion, or mv, is equal 



toc^ . For, by corol. 23, v = Ci/-^. 



2a •' 2ma 



Corol. 17. In the same case, mv is proportional to\/pLM, or to vt : and if 

 PL be given, either in the same, or different springs, mv is as v'm. 



This appears partly from the preceding corollary, where — — is a given quan- 

 tity ; and consequently mv is as ^viM ; and pl being given, mv is as Vta: and 

 partly from corol. 11 ; where t is as >/ — , and consequently rt is as \/plm. 



Corol. 28. In the same case, if - be given, either in the same, or in different 

 springs, the initial quantity of motion is as the length of the spring into the time 

 of bending it. For, by corol. 27, mv is as p<; and if p be as l, mv is as L.t. 



Corol. 29. If the quantity of motion mv bend a spring through its whole 

 length, and be consumed thereby, no other quantity of motion equal to the 

 former, as wm X -, will close the same spring, and be wholly consumed thereby. 



vol. IX. E 



