64 Mr maxwell, ON FARADAY'S LINES OF FORCE. 



The intensity measured along an element da of a curve 



e = la + mft + ny, 

 where Imn are the direction-cosines of the tangent. 



The integral feda 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/3 7 from equations (A) 



feda = JiXdo! + Ydy + Zdz) - p + C. 

 If, therefore {Xdx + Ydy + Zdz) is a complete differential, the value of feda for a closed curve 

 will vanish, and in all closed curves 



feda = f{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 



fedS, 

 where 



e = la + mb + nc, 



Imn being the direction-cosines of the normal, 



.*. fedS = ffadydz + ffbdzdx + ffcdxdy, 

 the integrations being effected over the given surface. When the surface is a closed one, then 

 we may find by integration by parts 



If we make 



da db dc 



— + T- + T" = 4'r|0 (C) 



dx dy dz 



fedS = 4 TT Jffpdxdydz, 

 where the integration on the right side of the equation is effected over every part of space 

 within the surface. In a large class of phenomena, including all cases of uniform currents, 

 the quantity p disappears. 



Magnetic Quantity and Intensity. 



From his study of the lines of magnetic force, Faraday has been led to the conclusion that 

 in the tubular surface* formed by a system of such lines, the quantity of magnetic induction 

 across any section of the tube is constant, and that the alteration of the character of these lines 

 in passing from one substance to another, is to be explained by a difference of inductive 

 capacity in the two substances, which is analogous to conductive power in the theory of 

 electric currents. 



• Exp. Res. 3271, definition of " Sphondyloid." 



