214 Prof. G. M. Minchin on the Magnetic 



D ; for, at points, P, close to the wire — ■ is nearly unity, and 



hence the fraction - — ^ is not negligible in comparison with 



L, unless, indeed, 8 a is many thousands of times greater 

 than c. 



Consider the magnetic force, Z E , just outside the wire at E. 



This is obtained by putting (/> = and m = c in -r- —tt ; thus, 

 omitting the factor i, 



Z E =-^-i(L + i)--^(12L+l), 



while by putting <j>=ir, m=e, we obtain the force, Z E , at B : 



Z B=--+-(L + i)- T7 f- 2 (12L + l), 

 c « loor v 



whereas the magnetic force at the centre, 0, of the circle is 



of the order — , and is therefore much less than the force 

 a 



close to the surface of the wire. 



Lines of Magnetic Force. — The forms, or approximate 

 equations, of the lines of magnetic force close to the wire 

 may be determined to the second order of small quantities in 

 like manner. Thus, in my previous paper on the Magnetic 

 Field of a Circular Current (Phil. Mag. April 1893, p. 356) 

 I have shown that if at any point in space in presence of a 

 current running in an infinitely thin circular filament G is 

 the vector potential due to the current, we shall have 



Gr . a = constant 



along the line of force, where a is the distance of the point 

 from the axis of the current (i. e., the perpendicular to its 

 plane drawn at its centre). In the case of a point so close 

 to a wire that the current through its cross-section must be 

 broken up into filaments (as in the previous calculation of ©), 

 the total vector potential at any point is \ Gc/S, and as a is 

 the same for all the filaments, the equation of a line of force 

 is 



J G . a . dS = constant. 



But (Phil. Mag. ibid.) 



Gc.x={{l + k' 2 )K-2E}p, 

 and the current-density in the filament through <iS being 



