Resistance and Inductance of a Helical Coil, 79 



satisfactory for all cases in which the pitch of winding is 

 very small. 



The present paper deals with the corresponding formulae 

 for a helical wire whose pitch is not very small, wound on a 

 cylinder whose radius is large compared with that of the 

 section of the wire, which is of course circular, and cannot he 

 treated as square in the manner applied by Cohen to the 

 other extreme case. A rigorous solution would be very 

 difficult, but it is shown that some very simple approximate 

 results are sufficient for practical purposes. The coil is to 

 be regarded as sufficiently long for the effects of the ends 

 to be neglected. A connexion between these formula? and 

 those of Cohen cannot be made without a consideration of 

 the difficult intermediate case in which the pitch is 

 moderately small. 



Choice of Coordinates. 

 Let the axis of the cylinder on which the wire is wound 

 be chosen as that of z in a Cartesian system. Defining the 

 central curve as the locus of the centres of all normal sections 

 of the wire, this curve will be a helix, and if a is the radius 

 of the cylinder, and 6 the twist in a plane perpendicular 

 to z of the radius in that plane measured from a line parallel 

 to a,; the coordinates of a point on the helix become 



,r = a cos 6, y — a sin #, z = a6 tan a. . . (1) 



where a is the angle of the helix. 



Let (p <f>) be polar coordinates in the plane of the normal 

 section at the point defined by 6. The initial line from 

 which cf> is to be measured is perpendicular to z and to the 

 tangent of tbe helix. Thus if z were vertical, the initial 

 line in any section would be the horizontal line. Now the 

 direction cosines, at the point #, of the tangent to the helix 

 are 



( — sin # cos a, cos # cos a, sin a). 



A line perpendicular to this and to the axis of z (0 01) is 

 readily shown to have a direction 



(cos0, sin 0, 0). 

 This is the line $ = in the section defined by 6. 



Let (X, /z, v) be the direction cosines of the radius vector 

 p in the section. It is at an angle cj> to the above line, and 

 is perpendicular to the central curve, so that 



A cos + fi sin 6 = cos </>, 

 — Xsin #cos a + yLt cos #ccs a+fsin a = 0, 

 with \ 2 + fjt, 2 + v 2 = 1. 



