356 Prof, G. M. Minchin on the Magnetic 



denoting the denominator by D. Now let yfr=7r— ©, and 

 let p 2 = (a + a) 2 + 7 2 , p 12 = (a - a) 2 + 7 2 , so that p = PB, p' = ¥A. 

 Then 





where 



a 2' 



Let a> = 20, and P = l— 4-; then, finally, 

 P 



G = — ||(K-E)-k}, 



where K and E are the complete elliptic integrals of the first 

 and second kinds with modulus k ; so that the quantity in 



PA 



brackets is a function of the ratio p— simply. 



Also, since p 2 — p /2 = 4aa, we have a= *-j— >and the quantity 



G . a which is constant along the line of force is given by the 

 equation 



G.a=^{2(K-E)-^ 2 K}. 



It is thus seen that at every point in space G is of the 

 form -/ ( - ) ; so that at all points on the surface for which 



" is a constant, the value of G will vary inversely as p. The 



surface for which - is constant is a sphere having its centre 



on the line BA produced and cutting the sphere having BA 

 for diameter orthogonally. If we assign a series of values to 



the ratio - , we obtain a series of spheres having their centres 



on BA and cutting the given sphere orthogonally, the radius 

 of each sphere of the series being, therefore, the length of a 

 tangent from its centre to the sphere described on BA; for, 

 given the base, BA, of a triangle, and the ratio of the sides, 

 the locus of the vertex is a circle whose diameter is the join 

 of the points which divide BA internally and externally in the 

 given ratio. The surface locus of the vertex is the sphere 

 generated by the revolution of this circle. 



