ON FARADAY'S LINES OF FORCE. 191 



Now the quantity of the current depends on the electro-motive force and 

 on the resistance of the medium. If the resistance of the medium be uniform 

 in all directions and equal to & 



a, = ^a a , & = A, y 2 = &A .......................... (B), 



but if the resistance be different in different directions, the law will be more 

 complicated. 



These quantities Oj, &, y, may be considered as representing the intensity 

 of the electric action in the directions of x, y, z. 



The intensity measured along an element da- of a curve is given by 



c = la + mfi + ny, 

 where Z, m, n are the direction-cosines of the tangent. 



The integral \tda- taken with respect to a given portion of a curve line, 

 represents the total intensity along that line. If the curve is a closed one, it 

 represents the total intensity of the electro-motive force in the closed curve. 



Substituting the values of a, $, y from equations (A) 

 \tdo- = \(Xdx + Ydy + Zdz) -p + C. 



If therefore (Xdx+ Ydy + Zdz) is a complete differential, the value of jecZo- for 

 a closed curve will vanish, and in all closed curves 



JecZo- = \(Xdx + Ydy + Zdz), 



the integration being effected along the curve, so that in a closed curve the 

 total intensity of the effective electro-motive force is equal to the total intensity 

 of the impressed electro-motive force. 



The total quantity of conduction through any surface is expressed by 



where 



e = la + mb + nc, 



I, TO, n being the direction-cosines of the normal, 



. '. \edS = \\adydz + \\bdzdx + \\cdxdy > 



the integrations being effected over the given surface. When the surface is a 

 closed one, then we may find by integration by parts 





