76 APPLICATION OF THE PRECEDING RESULTS 



there being less than the coercive force, and those where it 

 would be in motion ; moreover these parts would vary at every 

 instant, and the problem therefore become very intricate : were 

 we however to suppose the initial state so chosen, that the total 

 force to move any particle p within A, arising from its electric 

 state and exterior actions, was then just equal to the coercive 

 force /3 ; also, that the alteration in the exterior forces should 

 always be such, that if the electric fluid remained at rest during 

 the next instant, this total force should no where be less than 

 /3; the problem would become more easy, and still possess a 

 great degree of generality. For in this case, when the fluid is 

 moveable, the whole force tending to move any particle p within 

 A, will, at every instant, be exactly equal to the coercive force. 

 If therefore #, y, z represent the co-ordinates of p, and V the 

 value of the total potential function at any instant of time t, 

 arising from the electric state of the body and exterior forces, 

 we shall have the equation 



, f . 



+ -7- f.I-T-J ............ () 



\dy ) \dz ) 

 whose general integral may be thus constructed : 



Take the value of V arbitrarily over any surface whatever 

 S, plane or curved, and suppose three rectangular co-ordinates 

 w, w ', w", whose origin is at a point P on S: the axis of w 

 being a normal to S, and those of w', w", in its plane tangent. 



Then the values of -7, and -7-7, are known at the point P, and 

 dw dw 



dV 



the value of -7 will be determined by the equation 

 dw 



which is merely a transformation of the above. 



Take now another point P,, whose co-ordinates referred to 



dV dV , dV , , i r T 



these axes are -7- , -7, and -7-77 , and draw a ri^ht line L 

 dw dw dw 



