224 Trans. Acad. Sci. of St. Louis. 



If w = m' in equation (35), 



cos w' = cos CO = cos ( TT — (O) 



or 



a>' = TT — ft) • . . . ( '^6 ) . 



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



If in this case also we assume that the rotating lines extend in 

 both directions from their axes, and that co and to' can have any 

 values from minus infinity to plus infinity, we will get results 

 analogous to those reached in case (a). It will be convenient 

 in the following discussion to designate by the word travel ( for 

 short, trav) the distance moved through by the slotted cross- 

 head while the crank moves through a corresponding angle. 

 This word ( travel) may be used as an angular function. Thus, 

 by trav 240°, is to be understood the whole distance traveled 

 by the crosshead in moving the crank from a position of 

 coincidence with AX (Fig. 7) to a position making an angle 

 240° with AX. Trav m = versia co when gj <^ tt. Equation 

 ( 24) when written in the form 



m trav &> — m' trav oo' = m versin a (37) 



is the equation of a curve which may have a number of 

 loops and infinite branches, depending upon the relative values 

 of m and m'j and equation (25) when written in the form 



m trav oo ■}- m' trav co' = m versin a (38) 



is -the equation of a curve which may have, depending upon 

 the relative values of m and m', a number of infinite 

 branches, of which some pass through A, and the rest 

 through A' . 



In order to investigate the curve represented by equation 

 (37) for asymptotes, put for the first position of parallelism 



ftj' = ft), or trav &)' = trav co, 



for the 2"d, 



G>' = ft) + TT, or trav co' = trav co + s, 



