HELMHOLTZ ON THE CONSERVATION OF FORCE. 125 



letter d, are supposed to be fixed, so that q^ is constantly =0; 

 the form -of the law will then be 



S [<j>atdrab'] + 2 [^„d^r J = - 2 [^^.^^(g^,)] . . . (5) 



It remains to be shown in what relation the principle of the 

 conservation of force stands to the most general law of statics, 

 the so-called principle of virtual velocities. This follows imme- 

 diately from equations (3) and (5). If equilibrium is to set in 

 when a certain arrangement of the points ma takes place, that is, 

 if in case these points come to rest, hence 5'a = 0, they remain at 

 rest, hence dqa=0, it follows from equation (3), 



^ 2[<^a6flfr«J=0; . / (6) 



or in case that forces act upon them from points m^ without the 

 system, by equation (5), 



S[(^„At/r«,]+5;[<^arf^rJ=0 (7) 



In these equations under dr are understood alterations of 

 distance consequent on the small displacements of the point rria, 

 which are permitted by the conditions of the system. We have 

 seen, in the former deductions, that an increase of vis viva, hence 

 a transition from rest to motion, can only be effected by an ex- 

 penditure of tension ; in correspondence with this, the last 

 equations declare that in cases where in no single one of the 

 possible directions of motion tension in the first moment is con- 

 sumed, the system once at rest must remain so. 



It is known that all the laws of statics may be deduced from 

 the above equations. The most important consequence as regards 

 the nature of the acting forces is this : instead of the arbitrary 

 small displacements of the points m, let us suppose such intro- 

 duced as might take place were the system in itself firmly united, 

 so that in equation (7) every rfr„4 = 0, it follows singly, 

 2[(/)„6fl?raJ=0, and 

 ^[<l)abdrab]=0. 



Then the exterior, as well as the interior forces, must satisfy 

 among themselves the conditions of equilibrium. Hence, if any 

 system whatever of natural bodies be brought by the action of 

 exteiior forces into a certain position of equilibrium, the equi- 



