123 
of Edinburgh, Session 1862 - 3 . 
Newton with reference to machines and their visible motions only, 
but now extended to all the phenomena of physical science) con- 
sists the “ Conservation of Energy.” 
It would be easy to give the general investigation, but for an 
elementary lecture like this a very simple example will suffice, — 
the case of a particle moving in a straight line, and acted on by a 
force whose direction coincides with the line of motion. 
If s be the space passed over in time t, and v the velocity, geome- 
trical ideas lead at once to 
ds 
v = -— 
dt 
The second law of motion (above) gives, if be the force, and m 
the mass of the particle, 
F=iri^ ‘ ( 2 ) 
dt ■ ^ 
This is the ordinary equation of motion. 
But Newton’s second form of action and reaction, as connected by 
ds 
the third law, gives at once for the action of the force and for 
the reaction of the particle due to acceleration we have multi- 
plied by v. Hence 
77tC?s dv 
jj =mv—. 
dt dt 
.(3) 
which, as we see by (1), is merely the equation (2), with the 
additional factor v, or in each member. 
dt 
While the integrated form of (2) is 
showing that the momentum, or quantity of motion is increased in 
any interval by the product of that interval by the average value 
of the force, the integral form of (3) is 
( 3 )' 
