VOL. XLIII.] PHILOSOPHICAL TRANSACTIONS. 23 



equal to the right sine of the arch gf, that arch becomes a quadrant, and is 

 But / being eqiialto B = ^-5^, ■i;=v'2Si; therefore; = 1' X j^j X 



2ml ., 7H , Ml, ,;; », "^ / m/ 



^-- 'Ort. = 1" X 2 v^-27X= ^ X 2 ^2^^- 



Corol. 11. In the same case, the time of bending the spring is proportional 

 to */ — , or to \/ — ; and if - be given, t will be as -/ m ; and if both - and m be 



p 3 « p O p 



given, t will always be the same, whatever be the original velocity, or through 

 whatever space the spring be bent. 



Corol. 12. If the motion of the body cease when the spring is bent through 

 any space /, the product of the initial velocity, and the time of bending the 

 spring, or vt, is equal to l' X ^ ; and is proportional to /, the space through 

 which the spring is bent. 



For, by corol. 8, v = c//^"^' ^"^' by corol. 9, t = 1" x^y^^; there- 



tncl , MLP ,/ tficl 1 , /' 1 



fore vt = 1" X T-v^ — = 1' X :;— ; and, as 1', m, g, and a, are given 



quantities, vt is as /. 



Hence, any two of the three quantities, v, t, and /, being given, the other is 

 readily determined. 



Corol. 13. In the same case, the initial quantity of motion, or mv, is equal 



to c/v^_, or toe/— . 



For,bycorol.8,v=c/^-''-^ =c/^; therefore mv = c//£=c/^^. 



Corol. 14. In the same case, mv is proportional to 

 /y/— , or to '/plu, or to — , or to pt : and if- be given, mv is as 1\/m, or as It. 



For, in the preceding corol. 

 -^ is a given quantity; and by corol. 11, t is as^ — = ^Z— . 



Corol. 15. If the quantity of motion mv bend a spring of the strength p, and 

 length L, through the space /, and be wholly consumed thereby, no different 

 quantity of motion equal to the former, as wm X -, will bend the same spring 

 through the same space, and be wholly consumed thereby. 



For, by the preceding corol. if the spring be bent through the space /, and 

 each of these quantities of motion be consumed thereby ; / \/m : / ^nu :: mv : wm 

 X -. But MV = 7ZM X -; and therefore Ia/m = l\/^ nu, or 1 = n, and m = nu, 



' H n 



^nd V = -. Therefore the quantity of motion r?M X - is not equal to mv, but 

 is composed of an equal mass, and an equal velocity. 



>- 



iF 



