140 A COMPARISON OF THE ELECTRIC UNITS 



Let the direction of propagation be taken as the axis of z, and let all 

 the quant ities be functions of z and of t the time; that is, let every portion 

 of any plane perpendicular to z be in the same condition at the same instant. 



Let us also suppose that the magnetic force is in the direction of the 

 ^xyi of y, and let ft be the magnetic intensity in that direction at any point. 



Let the closed curve of Theorem A consist of a parallelogram in the 

 plane yz, two of whose sides are 6 along the axis of y, and z along the axis 

 of :. The integral of the magnetic intensity taken round this parallelogram is 

 b(ft, ft), where ft, is the value of ft at the origin. 



Now let p be the quantity of electric current in the direction of x per 

 unit of area taken at any point, then the whole current through the parallelo- 

 gram will be 



I bpdz, 



and we have by (A), 



l(ft t -ft) = 47r Plpdz. 



If we divide by b and differentiate with respect to z, we find 



Let us next consider a parallelogram in the plane of xz, two of whose 

 sides are a along the axis of x, and z along the axis of z. 



If' P is the electromotive force per unit of length in the direction of x, 

 then the total electromotive force round this parallelogram is a(P P ). 



If p. is the coefficient of magnetic induction, then the number of lines of 

 force embraced by this parallelogram will be 



and since by (B) the total electromotive force is equal to the rate of dimi- 

 nution of the number of lines in unit of time, 



Dividing by a and differentiating with respect to z, we find 



dft 



