344 THE MECHANICAL EQUIVALENT OF HEAT. 



bodies. In many cases falling-force is measured with suffi- 

 cient accuracy by the product of the raised weight into its 

 height ; and the expressions " foot-pound," " kilogramme- 

 metre," " horse-power," and many others, are conventional 

 units for the measurement of this force, which have of late 

 come into general use, especially in practical mechanics. But 

 in order to find the exact quantitative expression for the mag- 

 nitude in question, we must consider (at least) two masses 

 existing at a determinate distance from each other, which ac- 

 quire motion by mutually approaching ; and we must investi- 

 gate the relation which exists between the conditions of the 

 motion, namely, the magnitude of the masses and their orig- 

 inal and final distance, and the amount of motion produced. 



It very remarkably happens that this relation is the sim- 

 plest conceivable ; for, according to Newton's law of gravita- 

 tion, the quantity of motion produced is directly proportional 

 to the masses and to the space through 'which they fall, but 

 inversely proportional to the distances of the centres of grav- 

 ity of the masses before and after the movement. That is, 

 if A and B are the two masses, c and c' the velocities which 

 they respectively acquire, and h and h f their original and final 

 distances apart, we have 



or in words, the falling-force is equal to the product of the 

 masses into the space fallen through divided by the two distances. 

 By help of this theorem, which, as will be easily seen, is 

 nothing but a more general and convenient expression of 

 Newton's law of gravitation,* the laws of the fall of bodies 



* Newton's formula relates to the particular case in which the two dis- 

 tances (the initial and the final distance) are equal, so that their product 

 becomes a square. In this case, however, both the space fallen through 

 and the velocity become nought ; and hence, when this expression has to 



