70 APPLICATION OF THE PRECEDING RESULTS 



is supposed to conduct electricity imperfectly, and which will, 

 moreover, be interesting, as it serves to illustrate the magnetic 

 phenomena, produced by the rotation of bodies under the in- 

 fluence of the earth's magnetism. 



If any solid body whatever of revolution, turn about its axis, 

 it is required to determine what will take place, when the matter 

 of this solid is not perfectly conducting, supposing it under the 

 influence of a constant electrical force, acting parallel to any 

 given right line fixed in space, the body being originally in a 

 natural state. 



Let /3 designate the coercive force of the body, which we 

 will suppose analogous to friction in its operation, so that as 

 long as the total force acting upon any particle within the body 

 is less than /?, its electrical state shall remain unchanged, but 

 when it begins to exceed /3, a change shall ensue. 



In the first place, suppose the constant electrical force, which 

 we will designate by &, to act in a direction parallel to a line 

 passing through the centre of the body, and perpendicular to 

 its axis of revolution ; and let us consider this line as the axis 

 of x, that of revolution being the axis of z, and y the other 

 rectangular co-ordinate of a point p, within the body and fixed 

 in space. Thus, if V be the value of the total potential function 

 for the same point jo, at any instant of time, arising from the 

 electricity of the body and the exterior force, 



bx + V 



will be the part due to the body itself at the same instant : since 

 bx is that due to the constant force J, acting in the direction 

 of x, and tending to increase it. If now we make 



z = r cos O t x = r sin cos r, y = r sin 6 sin -or ; 



the angle r being supposed to increase in the direction of the 

 body's revolution, the part due to the body itself becomes 



br sin 6 cos -or + V. 



Were we to suppose the value of the potential function V 

 given at any instant, we might find its value at the next instant, 



