12 ON POTENTIAL. 



The components of the force with reference to any given axis 

 possess the same property. For let a line MA be drawn through 

 the point M, making an angle with the perpendicular, and let da 

 denote the portion of this line between the two surfaces S and S'. 

 The component F a of the force parallel to the straight line is thus 

 expressed : 



//V 



F =Fcos0 = -V-cosl9. 

 dn 



The figure gives moreover, 



dn = da cos 6 

 from which we get, 



_^V dn_ = _<)V 

 a ~ dn da" la' 



Thus, the component of the force in any given direction is equal, 

 and of opposite sign, to the partial differential of the potential along 

 this direction. 



If we consider three rectangular axes, the components X, Y and 

 Z of the force parallel to the axes are, 



(4) 



from which we have, 

 (5) 



25. EQUILIBRIUM OF CONDUCTORS. In the interior of a con- 

 ductor in equilibrium the force is zero (11). Hence for the whole 

 surface of the conductors we have, 



dV dV 3V 



- = 0, -- = 0, = 0, 

 OX oy 02 



and consequently, 



V = constant. 



It follows from this, that the whole volume of a conductor 

 in equilibrium is at the same potential ; this is what may be 

 called a level volume. Its surface being then a level surface, the 



