Field of an Electrical Current. 221 



In the case of variable current-density the equation deter- 

 mine- e is 



mining e is 



m /t i *\ — it m o 



iL-^(L-l-X)cos€=iL -^(L -l-X ); 



and applying this to the same numerical case, we find that p r , 

 q, r f , s' are determined by the angles 



68° 49'; 98° 10'; 123° 4'; 152° 4', 

 while the position of t f becomes imaginary. This shows that 

 the supposition of variable density brings the lines of force 

 closer to the surface, B, of the wire — as we should expect 

 a priori. It is not considered necessary to draw a separate 

 figure for the case of variable density, since the forms of the 

 lines of force and the method of drawing are sufficiently 

 illustrated by the case of assumed constant density. 



Initial or rapidly alternating Currents. — The magnetic 

 potential and the forms of the lines of force will not be the 

 same when the current has become steady as they were in the 

 initial stage of the current, because, just at starting, the cur- 

 rent is confined to the surface of the wire. If we can assume 

 that when the current is entirely superficial its density (or 

 its infinitesimal depth below T the surface of the wire) is con- 

 stant, the magnetic potential at any point P and the constant 

 of the line of force can be obtained by subtracting from the 

 value of the potential in (43) its value when c is replaced by 

 c — Ac, and a similar subtraction from (59^). Thus, the mag- 

 netic potential is the right-hand side of (43) multiplied by i, 

 the total current in the cross- section. If 8 is the density of 

 this current, i = 7rc 2 $, and if q is the total superficial current 

 ( = 2-7rc8 . Ac), the magnetic potential becomes 



-^{(6L-5)«( 2+ £)}]. 



From (49) it appears that the constant of the line of force 

 due to the current in the whole cross-section is (59) multiplied 

 by i } or by 7rc 2 5. When this multiplication is effected, the 

 differentiation with respect to c is to be performed, as in the 

 case of the potential. 



Of course the preceding discussion applies to the case of a 

 circular vortex ring in a perfect fluid, the velocity of a particle 

 at any point of the fluid being the analogue of the magnetic 

 force. See Basset's ' Hydrodynamics,' vol. ii. chap. xiv. 

 Mr. Basset assumes that for a vortex ring the magnitude of 

 the cross-section is negligible, so that the concentration of 



OCT / 



