\ . (3) 



SO . Dr. J. W. Nicholson on the Effective 



Thus 



1-v 2 = X 2 + fi 2 = (X cos 6 + fM sin <9) 2 + (> cos 6—\ sin 6) 2 

 = cos 2 <f> + v 2 tan 2 a, 

 or v = + cos a sin </>. 

 We choose the positive sign, and deduce 



X = cos 6 cos <£ + sin a sin 6 sin <£, "| 



fju = sin 6cos<j> — sin a cos sin 0, S . . (2) 



v = cos a, sin J 



as the direction of the radius vector. 



The coordinates of: any point of space are therefore 



x = a cos 6 + pX 1 



= (a 4- p cos <£) cos + p sin a sin sin <£, 



y = a sin 6 + pfM 



= (a + p cos <£) sin — p sin a cos 6 sin <£, 



z = ad tan a + pv j 



= a0 tan a + p cos a sin <£ 7 



We shall call (p <£ 0) the helical coordinates of any point of 

 space. If 8s (which only involves 80) be an element of 

 length perpendicular to a normal section at the point (p, <j>, 0), 

 then the space elements 8s, 8p, p8(f> are mutually perpen- 

 dicular, and the surfaces p = const, are those of helical wires 

 with a common central curve. This system of coordinates 

 is therefore appropriate to problems in which surface con- 

 ditions must be satisfied at the boundaries of such wires. 



Property of Orthogonal Systems. 



In the problem here contemplated, in which an alternating 

 electromotive force Ee^* is applied to the terminals of a long- 

 helical coil, the electric and magnetic vectors in and near 

 the coil will not depend on 0, if the other portion of the 

 circuit is kept at a distance. Thus B/c)0 = O. 



Consider now a general orthogonal system in which the 

 components of all vectors are independent of one coordinate. 

 Denoting the three coordinates by (p (/> 0), let the corre- 

 sponding space elements along their instantaneous directions 

 of increase at any point be 



(ds x , ds 2 , ds 3 ) - {dp/pk d<t>/p 2 , ddjp z ), . . (4) 



Let (a j /3, 7), (x, y, z) be the components of magnetic and 







