210 Trains. Acad. Sci. of St. Louis. 



From this and equation (15) 



or 



AO X m' = ^'0' X m (16). 



If m = wi', equation ( 15) becomes 



«' = (7r — «) (17). 



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

 In this case (when m = m') and 0' 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 eo and to' 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 m\ and equation (2) or (4) represents a curve which may 

 have, depending upon the relative values of m and m', 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 b)'^ equation (3) or (1). 



Tilt 071 ' H 



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

 m' m^ 3 



a — 50° with AO^ and AA^ 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 ^(x as a tangent at -4, is traced. This 



part approaches the asymptote marked /. After a position 



