122 HELMHOLTZ ON THE CONSERVATION OF FORCE. 



and from equation (2) of the foregoing section, we have 

 md{q^) = — 2^{xdx ■\-ydy + zdz) ; hence 



or when Q and R, q and r represent corresponding tangential 

 velocities and distances, 



lmGi^-'\mq^=-f <f>dr (2) 



Let us regard this equation more closely ; we find at the left- 

 hand side the difference of the vires vivcB possessed by m at two 

 different distances. To understand the import of the quantity 



R 



/ (f>dr, let us suppose the intensities of (ft which belong to dif- 

 ferent points of the connecting line ma erected as ordinates at 

 these points, then the above quantity would denote the super- 

 ficial content of the space enclosed between the two ordinates r 

 and R. As this surface may be regarded as the sum of the infinite 

 number of ordinates which lie between r and R, it therefore 

 represents the sum of the intensities of the forces which act at 

 all distances between R and r. Calling the forces which tend 

 to move the point m^ before the motion has actually taken place, 

 tensions, in opposition to that w^hich in mechanics is named vis 



viva, then the quantity / <l>dr would be the sum of the tensions 



between the distances R and r, and the above law would be thus 

 expressed : — The increase of vis viva of a material point during 

 its motion under the influence of a central force is equal to the 

 sum of the tensions which correspond to the alteration of its 

 distance. 



Let us suppose the case of two points operated upon by an 

 attractive force, at the distance R ; by the action of the force 

 they will be drawn to less distances r, their velocity, and conse- 

 quently vis viva, will be increased ; but if they should be driven 

 to greater distances 7-, their vis viva must diminish and must 

 finally be quite consumed. We can therefore distinguish, in the 

 case of attractive forces, the sum of the tensions for the distances 



between r =0 and r = R, / <^dr, as those which yet remain, but 



