EQUATIONS OF CURRENTS. 551 



If, further, we observe that the product Sds represents an 

 element of volume, we shall have 



W = - (Tu + Gv + llw) dxdydz. 



Replacing finally the components u, u, w by their values taken 

 from equations (4), we get 



f f f F /c)Y dZ\ /c)Z <)X\ /<)X 7)Y\"1 

 W= F( 1 + G( r ) + H( r--r- ) \dxdydz. 



&TT I \ oz oy I \ ox oz I \vy vX J I 



J J J L. \ / \ / \ /J 



We may integrate by parts each of the terms, which give, for 

 instance, 



r C>Y r *F 



F dxdydz = Wdxdy - Y dxdydz. 

 Repeating the same operation for all the other terms, we get 



~i 







J 



^ , , , , X I \dxdydz. 







If we extend this expression to the whole space contained in a 

 very distant surface, which comprises the whole system of the 

 magnets and of the currents, the first integral relative to the surface 

 itself is null, for the components of the magnetic force are inversely 

 as the cube of the distance. On the other hand, replacing in the 

 second integral the terms comprised within the brackets by their 

 values drawn from equations (2), we get 



W= (X 2 + Y 2 + Z 2 )^X*> 



or, if <f> is the magnetic induction at a point that is to say, the 

 product of the force by /*, 



