4 CLAUSIUS ON THE MECHANICAL EQU IVALENT 



be thus expressed: the increase of via viva in the system during 

 any given time is equal to the quantity of mechanical work pro- 

 duced during the, same time in the system. 



The determination of the work may be much simplified in 

 particular cases \Yhich very often present themselves. 



For instance, suppose a portion of the given forces to consist 

 of attractions and repulsions which remain equal throughout 

 the whole time, these forces being either such as might be ex- 

 erted by other points upon the given ones, in which case the 

 former must remain motionless, or such as the given points may 

 exercise upon each other; let the intensity of each of these 

 forces depend solely on the distance, and not upon the position 

 which the attracting or repelling points may occupy ; they may 

 be any function whatever of the distance. The portion of the 

 total sum due to these forces will then be expressed by 



l.^[Xdx + Ydy + Zdz)', 

 and not only is this a complete differential, because all the quan- 

 tities which appear in it are functions of one and the same va- 

 riable t, but it also remains so if the single quantities oc^y, z-y 

 a?, ?/', <s', &c., as far as is permitted by the express conditions to 

 which the general motion of the system is subject, be regarded 

 as independent variables. From this it follows that the value of 

 the integral 



f'Z^[Xdx-\rYdy-VZdz) 

 will be completely determined when the first and last values of 

 oCy y, 2 ; a/, y\ m' are known, without its being necessary to know 

 anything more regarding the nature of the motions by which the 

 masses m, mH, &c. pass from one position to another. 



Further, to take a more special case ; let the exterior forces 

 proceed partly from a system of immoveable masses //., /^', \j}\ &c., 

 and let these forces be inversely proportional to the squares of 

 the distances, so that if p denote the distance of the two masses 

 m and /a from each other, the force acting between them is 



+ -g-, which is positive or negative according as the force is 



one of attraction or of repulsion. Then for this portion Xg of 

 the total sum we have 



/S2(Xrfa? + Y</y4-Z(/^) = 2;±^ + const., . . . (3) 

 •^ P 



