342 PARTICULAR CASES. 



The density at a point of the surface is given by the equation 



-. o 



y* g* a* t 



/ being the perpendicular let fall from the centre, on the tangent 

 plane at the point whose co-ordinates are x, y, and z. 



The total charge of a zone determined by two planes perpendicular 

 to the axis a and at a distance dx, is equal to the product of the 

 intensity I by the difference dS of the sections of the ellipsoid corres- 

 ponding to these two planes. At a distance x the section is bounded 

 by the ellipse 



the surface of which is 



\ 

 we have then 



and, consequently, the charge of the zone is 



2Trbcl 



= xax. 



The ratio ^ of the charge of the zone to its height, which may 



be defined as the linear density, in reference to the axis of magneti- 

 sation, is therefore proportional to the distance of this zone from 

 the centre of the ellipsoid. 



358. When the axes of the ellipsoid are unequal, the coefficients 

 L, M, N are given by the partial differentials of a definite elliptic 

 integral; for the complete calculation we must refer to special 

 treatises, and must limit ourselves to giving the results relative to 

 ellipsoids of revolution. In this case, in fact, the problem is more 

 simple, and the coefficients are expressed by means of the ordinary 

 functions. 



