156 Ox THE MAXIMUM OF MECHANIC POWERS, AND 



over. Or if the velocity of .v, after having passed 

 through any space in any given time, T be multi- 

 phcd by .V, then that product will evidently be the 

 momentum of .r, after having passed through that 

 ^pace : and therefore, by the well known method of 

 fluxions, the value of .r may be obtained under the 

 circumstances of a maximum : and this will apply 

 to all the foregoing cases. But to select one of the 

 most useful, let it be tliat in Prob. I/, where the 

 lever moves on its center of gravity, which it is 

 generally made to do when a power is applied at 

 one end to raise a weight at the other to a certain 

 height, and then return to repeat its stroke, and so 

 continue by the alternate acting and ceasing of the 

 power. Now, in the case alluded to, the accelera- 



tivc force of .v is as ^~, therefore .v v/-^t-^ will be 

 as the momentum of .v after being urged by the 

 force by which it v.'ould be carried through a space, 

 that should be to the space a body would be carried 



throughbygravityinthe same time, as -^^ to unity. 

 Hence, by making the fluxion of .v V^-j^ equal to 

 nothing, we shall have 2'Px—:3x\rXt--.v—.vX 

 PZ=^=o, and therefore .r=x/^I5lEf±Z±Iiil^ Or 



4 



if the velocity of .r, after having passed through 

 any space in any given time, T be multiplied by d\ 

 the momentum is obtained at the end of that time, 

 let the space passed over be what it will. Now in 

 the above case —^X 16-77 f-et, is the space which 

 .r would pass through in the first second of time: 

 hence as r\- T\- : ^Xl6-i-; "E^I^X iG^feet, 

 equal to the spac e that .r would pass over in the time 

 I; therefore \/ ''~^J^ X 16"-^- is the velocity at the 

 end of that time, and .r v/ '^""^'^ X l6-h—T\/ Ibir 



Xv/--7xr~ 5sthe momentum, which, by making the 

 fluxion equal to nothing, will give .v as before. 

 It Mill be unnecessary to give examples of all the 



foregoing 



