72 APPLICATION OF THE PRECEDING RESULTS 



which equation expresses that the total force to move any par- 

 ticle p, within the body, is just equal to ft, the coercive force. 

 Now if we can assume any value for V, satisfying the above, 

 and such, that it shall reproduce itself after the electricity 

 belonging to the new total potential function (Art. 7), is allowed 

 to find its equilibrium with the coercive force, it is evident this 

 will be the required value, since the rest of the electricity is 

 exactly in equilibrium with the exterior force Z>, and may there- 

 fore be here neglected. To be able to do this the more easily, 

 conceive two new axes X', Y f , in advance of the old ones Jf, Y, 

 and making the angle 7 with them ; then the value of the new 

 potential function, before given, becomes 



/, ,dV , dV\ 



V+dco. (by cos 7 + bx sin 7 + y' -p- - x -p J , 



which, by assuming V= fty', and determining 7 by the equa- 

 tion 



= b sin 7 ft, 



reduces itself to 



y (ft + b cosydco). 



Considering now the symmetrical distribution of the electricity 

 belonging to this potential function, with regard to the plane 

 whose equation is = y', it will be evident that, after the elec- 

 tricity has found its equilibrium, the value of V at this plane 

 must be equal to zero : a condition which, combined with the 

 partial differential equation before given, will serve to determine, 

 completely, the value of V at the next instant, and this value of 

 V will be 



V=fty. 



We thus see that the assumed value of V reproduces itself at 

 the end of the following instant, and is therefore the one required 

 belonging to the permanent state. 



If the body had been a perfect conductor, the value of V 

 would evidently have been equal to zero, seeing that it was sup- 

 posed originally in a natural state : that just found is therefore 

 due to the rotation combined with the coercive force, and we 



