120 HELMHOLTZ ON THE CONSERVATION OF FORCE. 



in the direction of the lines which unite them, and the intensity 

 of which depends only upon the distance. In mechanics such 

 forces are generally named central forces. Hence, conversely, 

 it follows that in all actions of natural bodies upon each other, 

 w here the above principle is capable of general application, even 

 to the ultimate particles of these bodies, such central forces 

 must be regarded as the simplest fundamental ones. 



Let us consider the case of a material point with the mass m, 

 which moves under the influence of several forces which are 

 united together in a fixed system A; by mechanics we are 

 enabled to determine the velocity and position of this point at 

 any given time. We should therefore regard the time t as pri- 

 mitive variable, and render dependent upon it, — the ordinates 

 37, y, ;2 of m in a system of coordinates, definite as regards A, 

 the tangential velocity g, the components of the latter parallel 



to the axes, «<= — , '^=-^9 ^=777? ^^^ finally the components of 



the acting forces 



V du ^^ dv rw dw 

 ■^^^-JJ^ Y=m-7-, Z=m-7:. 

 dt' dt^ dt 



Now according to our principle - m (f, and hence also g^ must 



be always the same when m occupies the same position relative 

 to A ; it is not therefore to be regarded merely as a function of 

 the primitive variable /, but also as a function of the coordinates 

 07, y, z only ; so that 



^<.-)=f-'^-f-'*4f- ■•■<■) 



As 5'2 _- ^2 _f. ^2 ^ yji^ ^g jj^^^g ^ ^^2j _ 2 udu + 2vdv + 2 wdw. 



Instead of u let us substitute its value ^, and instead of du its 



dt 



value -^, the corresponding values of v and w being also used, 



we have 



</(^2)^2X^^^2Y 2Z^^ 



m mm ^ ' 



As the equations (1) and (2) must hold good together for all 



values whatever of dx, dy, dz, it follows that 



d{q^) _ 2X d(q^) _ 2Y . d{q^) _ 2Z 



■ -5 , — anci — . 



ax m dy m dz m 



