EQUATIONS OF THE MAGNETIC FIELD. 547 



on the projections dx, dy> and dz, of the element ds. We shall 

 have then the equation 



the integral being extended to the whole circuit. 



On the other hand, if we consider any given surface S bounded 

 by a circuit, and we denote by X, Y, and Z the components of the 

 magnetic force at a point of the surface, and by a, ft, and y the 

 cosines of the angles of this force with the perpendicular, the total 

 flow of magnetic induction across the circuit is also expressed by 



the integral /* (Xa + Y/3 + Zy) dS, extended to the whole surface. 



567. In order to determine the relations between the electro- 

 motive force J and the magnetic induction, we shall calculate the 



value of the integral Jds cos e for an infinitely small rectangular 



circuit with its centre at the origin of the co-ordinates, perpen- 

 dicular to the x axis, and the sides of which are equal to dy 

 and dz. 



Let F , G , H be the values of the functions F, G, H at the 

 origin; we shall have, for the top side of the rectangle, 



and for the bottom side 



If we follow the contour in the direction in which the current 

 would be produced by the suppression of the field, the sum of 

 the two terms of the electromotive force corresponding to the 

 sides parallel to the y axis, is G 3 dy G 2 dy ; we get thus 



c)G 

 G-^dy - G 2 dy = dzdy. 



02 



We shall have, in like manner, for the two other sides, 



<)H 



- Hj dz + H 2 dz = -- dydz. 



N N 2 



