Boever — Geometrical Constructions of Lines of Force. 213 

 To investigate the curve for tangents at A put for the 



l)7r, 



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

 becomes 



CO, =«+^(n, — l)7r (19), 



in which o), is the special value of a> when line A'P coincides 

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

 tangents at A to two consecutively formed parts of the curve 



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



A to parts 1 and 2, or to parts 3 and 4, is — = -tt. _ is 



7n S 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** tangent w = 



2"«* *' O) = TT 



n,"' ** &) = (n,— 1) TT, 



in which n^ is an integer. For the n2"' tangent equation ( 1 ) 

 becomes 



mm ,^^ 



6);=_— ;« + —(71.,— 1) TT (20), 



in which oi\ is the special value of a>' when line AP coincides 

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



between two loops, consecutively formed, is — ^ tt. In this 



expression mir is the angle swept through by the rotating line 

 AP before a position of parallelism coincides with a pre- 



