202 Prof. Gr. M. Minchin on the Magnetic 



where & 2 =1— ^-, and II is the symbol for the complete 



r 



elliptic integral of the third kind, its parameter and modulus, 

 respectively, being the quantities included in the brackets 

 following II, according to the ordinary notation of such 

 integrals, viz., 



n ( W > *) = 77^ • 2 \ /i Tt • 2 ' 



J (1 +nsin^©)v 1 — rsira 



If P' lies anywhere in the plane of the circle and inside its 

 area, X2 = 27r; and if P 7 lies anywhere in the plane of the 

 circle and outside its area, £1 = 0. If P 7 is taken strictly on 

 the circumference of the circle, and if the circle is a strictly 

 Euclidian curve, i. e. something absolutely devoid of breadth, 

 H is necessarily indeterminate. It is well known that all the 

 surfaces of constant conical angle subtended by a circuit of 

 any form, plane or tortuous, contain the circuit as a bounding 

 edge, and that any two surfaces for which £1 = ^ and f2 = £2 2 

 are inclined to each other at the constant angle whose circular 

 measure is J (Oi — X2 2 ) a ^ all points on this common bounding 

 edge. 



When the circuit is not a Euclidian curve, but a wire, and 

 the point in space which we consider is near the surface of 

 the wire, as at P in fig. 1, it becomes necessary to take account 

 of the dimension of the cross-section of the wire, and the 

 conical angle subtended at P is the integrated result of divi- 

 ding the. normal cross-section made by a plane through P into 

 an indefinitely great number of indefinitely small elements of 

 area and breaking up the wire into a corresponding number 

 of circular filaments having these elements of area for cross- 

 sections, these circular filaments having all the same central 

 axis, OY, and their planes being, of course, all parallel. 



Our object now is to find the value of 12 when P is very 

 close to the wire both when the current is assumed to be of 

 constant density and when it is assumed to be of variable 

 density in the cross-section. Fig. 2 represents the normal 

 cross-section of the wire (supposed to be a circle) made by a 

 plane through P. 



Take any point, Q, in the cross- section, and at Q take an 

 indefinitely small element of area, r/S. If i is the total cur- 

 rent flowing through the cross-section of the wire, the current 



in the first case through d$ is i —g, where c is the radius of 



ire* 



the cross-section; and if £1 is the conical angle subtended 

 at P by the circular filament of the wire passing through dS, 



