Moever — Geometrical Constructions of Lines of Force. 213 

 To investigate the curve for tangents at A put for the 



in which n 1 is an integer. For the n* h tangent equation (1) 

 becomes 



«>, =« + ^-(«i — 1)«" ( 1<J )' 



in which co t is the special value of co when line -4'P coincides 



with A' A. Equation (19) shows that the angle between the 



tangents at A to two consecutively formed parts of the curve 



vn 7T 

 is . Thus, in Fig. 5, the angle between the tangents at 



m ° ° 



^4 to parts 1 and 2, or to parts 3 and 4, is — — -ir. — is 



m 8 m 



the angle between two adjacent tangents at A, and m is equal 



to the number of tangents at A. 



To investigate the curve for tangents at A' put for the 



1) 7T, 



in which n 2 is an integer. For the n, th tangent equation ( 1 ) 

 becomes 



m ni 

 ©;=_— « + — (ra 2 — 1) 7T (20), 



in which &>^ is the special value of &>' when line" ^.P coincides 

 with AA . Equation (20) shows that the angle at A' 



between two loops, consecutively formed, is — . it. In this 



m 



expression mir is the angle swept through by the rotating line 



A'P before a position of parallelism coincides with a pre- 



