450 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. 



different, so long as the attracted point lies outside them. These 

 shells may be considered to form a continuous body, which is, 

 being divided into shells, lamellarly magnetized, the potential of 

 magnetization <f> being equal to the current-function ^ ( 124). 



The magnetic potential Q is accordingly at outside points, by 

 124(n), 



HH6 



But since the form of the magnetic shells is indifferent, as long 

 as their edges are of the given shape, we may consider them all 



deformed so as to coincide with 

 the current-sheet, as is illustrated 

 in Fig. 88. The shells overlap 

 each other continuously, so that 

 there are more shells laid on the 



sheet the greater the values of " V F. 

 Fm. 88. 



As we cross the sheet, the poten- 

 tial fl is discontinuous, as in the case of a single magnetic shell. 

 As in that case also, the normal component of the magnetic force, 

 being continuous for all the shells, is continuous on crossing 

 the sheet. The tangential component in the direction of the 

 lines of flow is also continuous, but, as we found at the end of 228, 

 the component perpendicular to them experiences a discontinuity 

 equal to 4?r times the current-density, that is 4>7rdW fin*. This may 

 also be very simply obtained by taking the line-integral of magnetic 

 force around any circuit composed of two infinitely near portions 

 lying on opposite sides of the current-sheet and coinciding with 

 an electrical equipotential line, the integral being equal to 4?r 

 times the difference in the values of the current-function at the 

 two points where the circuit cuts the sheet. 



231. Examples. Coefficients of Induction. Toroidal 

 and straight coils. We shall now calculate the energy due to 

 currents in a few simple cases. The coefficients of the half-squares 

 and products of the current-strengths in the expression for the 

 electrokinetic energy, are called, for reasons to be explained in the 

 next chapter, coefficients of induction, or more briefly, inductances, 

 distinguishing coefficients of half-squares by the name self-in- 

 ductaiice, coefficients of products by the name mutual inductance. 



