MOVING WITH DIFFERENT VELOCITIES. 367 
spring must be communicated to the bodies A and B, against which they act re- 
spectively ; and as these forces are to each other as 1 to 10, so the forces of A and B 
must be as 1 to 10. But the velocities of A and B are to each other the same that 
they were in the first experiment. Hence it follows, that the forces produced in A and 
B by the unbending of the spring in the first experiment were not equal, but that B 
received 10 parts of the force and A 1 part; and this distribution corresponds exactly 
with the product of the mass into the square of its velocity. 
' :, For the mass of A = 10 and its velocity — 1; then 10 X 1? = 10 for the force of 
A; and the mass of B — 1 and its velocity = 10 ; then 1 X 10? — 100 for the force of B. 
We shall find, moreover, that the forces of A and B, under the velocities produced 
by the action of the spring, are to each other inversely as the masses of A and B; that 
is, at the instant that they leave the spring, 
Force of A: Force of B :: Mass of B : Mass of A. 
Or, the force of B is as much greater than the force of A as the mass of A is greater 
than the mass of B. Again, if we neglect entirely the masses of A and B, or the quan- 
tity of the mass in each, we shall find that their forces are to each other as their veloci- 
ties directly, and not as the squares of their velocities. Thus, we have found the 
force of B at the instant that it leaves the spring to be ten times as great as that of A. 
This was found by multiplying the mass of each into the square of its velocity. But 
the velocity of B is ten times as great as that of A. If therefore we neglect the masses, 
as factors, with both bodies, and take the force of each, not as the square of its velocity, 
but as its velocity simply, we have the same'comparative results; that is, 
Force of B: Force of A :: Velocity of B : Velocity of A.* 
This last statement is not given as favoring the Newtonian measure of force. For 
the mass of the body is, in that, always taken as an element or factor of the force, and 
it can be struck out only in a case like the one here stated ; where, on striking it out, 
the value of another factor of the force, namely the velocity, is evolved from the square 
to its simple power. 
It cannot fail to be understood, that whenever two bodies, as A and B in the pre- 
ceding example, receive the action of elasticity impelling them in opposite directions, 
the neutral point, as at a, Fig. 2, will be placed so as to divide the elastic force into 
two parts, having the same proportion to each other that the parts into which a line 
connecting the centres of gravity of the two bodies will be divided by their common 
- * Indeed, this amounts to no more than dividing the two sides of an equation by the same number. 
