210 Trans. Acad. iSci. of St. Louis. 



From this and equation (15) 



/ ^* \ 



sin —, (tt — Q)) 



r' sin ( TT — co) 



In the limit, when co = "tt, 



r A O' m 



? ^ A a ^ 1^' 



or 



AO' X m' = AO'Xni (16). 



If m = W2', equation ( 15) becomes 



ft)' = f TT ft)) (17). 



This is the equation of the perpendicular bisector to A' A. 

 In this case (when m — m') O and O' coincide and are at the 

 middle point of A A. 



If we assume that the rotating lines extend in both direc- 

 tions from their axes, and that w and &>' can have any values 

 from plus infinity to minus infinity, equation (1) or (3) rep- 

 resents a curve which may have a number of loops and 

 infinite branches, depending upon the relative values of m 

 and wi', and equation (2) or (4) represents a curve which may 

 have, depending upon the relative values of m and w', a num- 

 ber of infinite branches, of which some pass through A and 

 the rest through A ., but none of which pass through both A 

 and A. 



Consider first the case in which the lines rotate in the 

 same direction, which is represented by equation (3) or (1). 



In Fis'. 5, in which — = — ? = — numerically, AG^ at angle 



im! wij 3 



a — 50^ with AO., and A A., coinciding with AO, are the 



initial positions of the rotating lines AP and AP. As 



these lines rotate in a counter clockwise direction, the part 



marked 1, and having ^6r as a tangent at ^, is traced. This 



part approaches the asymptote marked /. After a position 



