980 TRANSACTIONS OF THE AMERICAN INSTITUTE. 



space. A retardation, -which is simple the action of a force in the 

 opposite direction to that of the moving bod3^ causes in it a vir- 

 tual motion in the opposite direction of constantly increasing velo- 

 city, until at last the two motions become equal, and the body is 

 brought to rest for an instant. 



On the other hand a body, whether at rest or in motion, if acted 

 upon by an accelerating force for a definite period of time, moves 

 through a greater space than it otherwise would, and the excess of 

 distance so traversed is called the " effective space," and becomes 

 the measure of " vis-viva," " living force," or " power of perform- 

 ing work," as it is variously called. 



It is not difficult to see that the distance which a body will tra- 

 verse, when accelerated from zero by a uniformly acting force, or 

 retarded by such a force to zero, is proportional to the square of 

 the terminal velocity in one case and of the initial velocity in the 

 other. 



For, we may suppose the units of time and of space as minute 

 as we please, and such that during each unit of time, the increase 

 of velocity is a unit of space. During the first unit of time the 

 inean velocity will be one-half the unit of space, the initial velocity 

 being'O, and the terminal velocity 1. For the second unit of time 

 the mean velocity will be 1^ ; for the third 2^, etc. ; and since 

 these mean velocities measure the distance traversed during each 

 unit of time, the spaces traversed during a succession of such 

 units are as ^, 1^, 2^, 3|, etc., or clearing of fractions, as 1, 3, 5, 

 7, 9, etc. Comparing the total spaces traversed, we find them, as 

 1, l-]-3=:4, 4-1-5=9, 9-f7=16, etc., or a's the squares of the 

 times or of the terminal velocities. In the case of a retardation 

 to zero, exactly the reverse takes place, and the spaces are pro- 

 portional to the squares of the initial velocities. 



The force, therefore, acquired by a falling body is capable of 

 raising the same body to the exact height from which it fell. A, 

 body having twice the velocity of another, is capable of raising a 

 weight four times as high, or it will raise four times as much 

 weight to the same height as the other body. 



It follows from the principle of the relativity of motion, already 

 stated, that when a change of condition is effected, the effective 

 space is to be computed in the same way as if the body were at 

 " rest at the commencement or termination of the change of condi- 

 tion. To determine it therefore, we have only to multiply the 

 amount of the change in velocity by the time of its accomplish- 



