HELMHOLTZ ON THE CONSERVATION OF FORCE. 121 



But if (f- is a function of x, y, and z merely, it follows that X, Y, 

 and Z, that is, the direction and magnitude of the acting forces, 

 are purely functions of the position of m in respect to A. 



Let us now imagine, instead of the system A, a single material 

 point «, it follows from what has been just proved, that the 

 direction and magnitude of the force exerted by a upon m is only 

 affected by the position which m occupies with regard to a. But 

 the only circumstance, as regards position, that can affect the 

 action between the two points is the distance ma; the law, 

 therefore, in this case would require to be so modified, that the 

 direction and magnitude of the force must be functions of the 

 said distance, which we shall name r. Let us suppose the co- 

 ordinates referred to any system of axes whatever whose origin 

 lies in c, we have then 



md[q^) = 2Xdx-[-2Ydy + 2Zdz=:0, ... (3) 

 as often as 



d[r'^) = 2xdx + 2ydy + 2zdz = 



that is, as oflen as 



xdx + ydy 



dz=- 5 



z 



setting this value in equation (3), we obtain 



/'X-^Zy^+ (Y-|zyj/=0 

 for any values whatever of dx and dy ; hence also singly 



X = -ZandY=^Z, 



z z 



that is to say, the resultant must be directed towards the origin 

 of coordinates, or towards the point a. 



Hence in systems to which the principle of the conservation 

 of force can be applied, in all its generality, the elementary forces 

 of the material points must be central forces. 



II. The principle of the Conservation of Force. 



We will now give the law for the cases where the central 

 forces act, a still more general expression. 



Let <\) be the intensity of the force which acts in the direction 

 of r, which is to be regarded as positive when it attracts, and as 

 negative when it repels, then we have 



