Field of a Circular Current. 363 



spherical radius vector op drawn to any point, p, of the curve, 

 and cj) is the angle between the radius op and any fixed arc, 

 oa, drawn at o. If, as said, the pole o is inside the area, <£ 

 goes from o to 2-7T ; but if o is outside the curve, the area has 

 a different expression, viz.: — 



J cos 0dcj), 



the longitude angle (f> obviously starting and ending with a 

 zero value. If o is on the curve, the expression for the area 

 is again different. 



In calculating the area of the above ellipse it would be 

 natural to choose for pole (o) the point n in which the sphere 

 is cut by the line PN ; but this leads to difficulties when the 

 position of P is such that n falls on the ellipse. This will 

 happen when P is on any perpendicular to the plane of the 

 circle of the current drawn at any point on its circumference ; 

 and, moreover, the choosing of n for pole will lead to expres- 

 sions for the conical angle which present its values in forms 

 which are apparently discontinuous for points P which project 

 inside and outside the area of the given circle BMA1. Such 

 discontinuity must not exist, and to get rid of it from the 

 expressions requires troublesome transformations of elliptic 

 integrals of the third kind. 



We must, then, choose for pole a point which is always 

 inside the spherical ellipse. The simplest point is the point 

 o (fig. 4) , in which the sphere is cut by the line PO which 

 joins P to the centre, 0, of the given circle. This point is, 

 of course, always inside the ellipse. 



Let, then, Q be any point on the given circuit, and p the 

 point in which PQ cuts the ellipse. Taking for the fixed 

 plane of longitude through o the plane baP, or BAP, and 

 denoting the angle poa by cf>, the area of the ellipse is 



f (1— cos op) d(j>, i. e., 2tt— f cos op. d(j>. 



Denoting, as before, the position of Q by. the angle ty, or 

 QOA, we easily find, if PN=^, PO = r, 0N = «, 



^^ z 2 + x 2 sm 2 yjr ' V> 



r 2 — aajcosyjr 



C0S ° P = r~7W+oJ^a = ^G = o7^ i 

 Hence 



C" ja. 9 f" r 2 - ax cos^ dyjr 



\ cos op .d<p~zz I t> 1 — - _. — ,/. ,. , . 



Jo Jo v^ + fl 2 - 2awcosyjr r + ^sin~x/r 



