118 OUTLINES OF NATURAL PHILOSOPHY. 



ax 



the velocity lost by A is g ; and the veloci- 

 ty gained by B is g -f x ; now the momenta of these 



must be equal by 120, that is, A a (g ~\ 



b / 



= B b (g -f x) ; and hence <r, or the velocity of B, 

 is found, and the half of it is the space over which 

 B is raised in one second. 



Hence the time in which B will be raised over a 

 given space h, being equal to as many seconds as h 

 contains the above expression, is = 



A a 2 + B 6 2 2 A 



s\ 



(A a B b) b g * 



It is evident, that if A is given, as also B and b, and 

 if the length of the arm a be alone supposed vari- 

 able, the time thus computed must admit of a mi- 

 nimum, as it would be infinite if a were so short, 

 that A a B b ; and it would also be infinite, if a 

 were infinitely long. There must be a value of #, 

 therefore, that will give the time of ascent the 

 least, or the angular velocity of the lever the great- 

 est ; and it will be found by the preceding for- 

 mula- 



4 



187- The same things, therefore, being suppo- 

 sed as in the last proposition, the angular velocity 



of 



