Field of a Circular Current. 359 



whereas above we have 



2 \/T^k 2 

 r = — p— .«. 



Of course (as stated in a note by Clerk Maxwell) the elliptic 

 integrals depending on the one modulus can be transformed 

 into elliptic integrals depending on the other ; and in this 

 case the transformation is the well-known one of Lagrange. 

 But the constructions for the points in which any given line 

 of force cuts the series of circles will not be the same in both 

 cases — those of Clerk Maxwell depending on a series of right 

 lines perpendicular to BA, and those above indicated de- 

 pending on a series of radial distances from B. 



When we propose to draw the line of constant vector 

 potential through any point, P, which lies on a circle whose 

 constant is Q , let PB be p ; then the point, K, in which this 

 line meets any other circle, whose constant is Q, is found from 

 the relation 



Q 



where p = BR. 



This latter method has a certain advantage for the eye, 

 inasmuch as it enables us to see readily those circles of the 

 series outside which the line of constant vector potential 

 through any proposed point lies. 



Consider now the lines of force. With the above value of 

 Q, the quantity which is constant along a line of force is 

 p . k 2 Q, so that on each of the above circles in fig, 2 we must now 

 mark the number k 2 Q. Denote this by Q'. Then the above 

 relation for points on the same line of constant vector potential 

 becomes for the lines of force 



n-n Qo'- 

 P-P0QT> 



and the construction proceeds in the same way. The con- 

 stants, Q', for the above series of circles, beginning at the 

 innermost, are : — 



•4841; -4301; -3775; -3396; -2782; -2376; -1954; '1727. 



! The values of the Q's diminish outwards for the circles ; so 

 that if we consider the line of vector potential at any point, 

 S, suppose, which is such that SB is greater than the distance 

 from B of the point along AC in which any circle interior to 

 that passing through 8 cuts the line BAC, it is at once obvious 

 that the line of vector potential which belongs to S is wholly 

 outside all such circles. The numerical values of Q for the 



