358 Prof. Gr. M. Mincliin on the Magnetic 



the length PB such that 



PB = ^'. a. 



Let PT be any circle of the co-orthogonal series cutting BA 

 at n and m. Then for this circle 



p' _ An _ mA . 

 p Bn mB 



and if this ratio is denoted by s, it is well known that 



CA_ 2 

 CB~~ 5 ' 



where C is the centre of the circle. Now the modulus, k, of 

 the elliptic integrals which belongs to the circle niPn is 



(l — prpj? *'.*• k 2 = l — s 2 ; hence 



k ~BC ? 



or the square of the modulus is inversely proportional to the 

 distance, BC, of the centre of the circle from B. 



The circles employed by Clerk Maxwell in drawing the 

 lines of force can be easily shown to be this co-orthogonal 

 system whose centres are ranged along BA produced. For, 

 his rule is to assign a series of values to 6, and construct 

 a series of circles whose centres lie on BA, the radius of 



each being ^ (cosec 6 — sin 0), while the distance of its centre 



from is ~ (cosec 6 + sin 6) ; the modulus belonging to this 



circle is sin 6. For the series of circles he then calculates 

 the values of the expression (constant for each circle) 



y= — ^2, and the point on each circle which lies on any 



assigned line of force is found by drawing a certain right 

 line perpendicular to BA. It is at once found that this 

 series of circles is precisely the co-orthogonal system above 

 described ; but Clerk Maxwell's modulus is not the same 



pi 

 function of the ratio -, or of the radius of the circle selected, 



P . 



as that adopted above ; for, with Clerk Maxwell, if r is the 

 radius of any circle of the series and k the corresponding 

 modulus, 



-!G-> 



