212 Prof. Gr. M. Minchin on the Magnetic Field 



the perpendicular at B to AB, and drawing perpendiculars at 

 m,n,p, ... to Am, An, Ap, . . . ; the points of intersection of 

 AB produced with these perpendiculars are the centres of the 

 circles. If C is the centre of the orthogonal circle through 



m. we know that the constant, 1 — ^-5 , on this circle is -m 1 

 ? ' p 2 ' AC 



i. e., cos 2 mAB. Hence if mAB=/3, we have 



k=cosj3; F = sin/5 (20) 



The field due to the plate AB is most readily mapped out by 

 describing a large number of very close circles of the ortho- 

 gonal system for a regular gradation of the values raAB, 

 nAB, pAB, . . . of /3, drawing a line BP in the assigned 

 direction 6, and from Legendre's tables of Elliptic Integrals 

 taking out the values of K, E, Kg, E' fl . 



The properties of the orthogonal circles lead to some simple 

 results with regard to potentials. Thus, if any line, AP, is 

 drawn from A cutting any circle of the series in P and P', 

 the lines joining P and P' to B are equally inclined to AB, 

 i.e., Z.ABP = 7T-0. 



Now if in A e we put 7r— 6 for 6, we have, in virtue of 

 Legendre's relation between complete complementary inte- 

 grals, A n _ e =7r-A e , i. e., 



A .-*+ A e = *- (21) 



Hence, from (18), if 12, Q,' are the conical angles subtended 

 at P, P' respectively by the circle (or plate) AB, we have 

 the remarkable relation 



Il + a' = 27r-4^Ksin(9 (22) 



Again, if V, V are the potentials at P, P' due to the plate, 

 we have from (15) 



-^ + -^r, = 2m(2E-7r&' sin 6), . . . (23) 



a result which enables us to lay down the field at all points 

 to the right of the perpendicular Bp when the field to the 

 left of Bp is known. 



Supposing now that instead of a single wire of diameter 

 AB, we have a series of wires forming a coil contained be- 

 tween the diameter AB and the diameter ST, i. e., the breadth 

 of the coil is BT or AS ; then in calculating the potential at 

 P we shall have to find the potentials due to a series of cir- 

 cular plates, each of surface-density m, and to add these 

 potentials together. But observe that the potential at P due 



