229, 230] ELECTROMAGNETISM. 449 



4?r 4?r \dy 0*J' 



^-^ AG =-^ A '---i' 



which would produce the same magnetic effect. It was for this 

 reason that Ampere was led to the hypothesis that all magnetism 

 was due to currents of electricity circulating about the molecules 

 of matter. 



The above formulae all refer to currents and magnets placed in 

 a homogeneous medium, and as has been already seen, lose all 

 their simplicity when the inductivity varies. For although we 

 may still calculate the vector potentials due to the induced 

 magnetization, the process will be complicated, and in general 

 impracticable. For this reason, and because both scalar and 

 vector potentials are quantities whose physical significance is 

 much less apparent than that of the strength of the field, 

 Heaviside and Hertz have been led to avoid the employment of 

 potentials, and to deal directly with the electrical and magnetic 

 fields. We have however introduced the vector-potentials here 

 on account of their important mathematical relations, and the 

 fact that they have been so much used by the highest authorities. 



230. Magnetic Field due to Current- Sheet. We have 

 found that in a current-sheet the amount of electricity that flows 

 in unit time across a curve connecting any two points in the sheet 

 is equal to the difference of the current-function SP at those two 

 points. This quantity is the same whatever the curve connecting 

 them, unless there is an electrode lying between. We shall sup- 

 pose that a sheet has no electrodes, so that the current flows in 

 closed circuits in the sheet. We may find the magnetic field of 

 such a sheet, at points not lying in the sheet, by the consideration 

 that the strip of the sheet bounded by the curves NP" = const, and 

 ^ -I- dW = const., dW being a constant difference in the values of 

 the current-function for the two curves, is equivalent to a linear 

 current of strength dW. Such a current, by 210, is equivalent 

 to a magnetic shell of strength dW. The whole current sheet 

 may therefore be replaced by an infinite series of magnetic shells, 

 whose edges only are given, the form of the shells being in- 

 w. E. 29 



