296 Trans. Acad. Set. oj St. Louis. 



the curves represented by equation (43a). The second form 

 of equation (45) suggests an easy method of construction. 



The curve represented by equation (43) is tangent to some 

 line of force at its point of inflection. 



The slope of a normal to a line of force at the point 

 {x\y') is 



^j, 1 mx' 



^ iira {x'-^ -\- y"^y — my' 



Hence the equation of a normal is 



TYlx' 



y — y' = i (^ — ^') (46). 



2170 {x"^-\-7j"^y — my' 



The condition that the point (x', y') shall be a point of in- 



rti 



flection is y'-\/x''^ ~^y'^ = ~o — • ^^^ this condition equation 



(46) becomes 



y' 

 y = y' + -' {x — x'), 



from which y = when cc = 0. This shows that normals ta 

 lines of force at their points of inflection pass through the 

 origin 0. 



If in this case also we assume that the rotating line extends 

 in both directions from its axis and that co can have any value 

 from minus infinity to plus infinity, we 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. 

 Trav CO = versin w when oj <^7r.* Equation (25) written in 

 the form 



m trav co — irax"^ = K 



* See Trans. Acad. Sci. of St. Louis. Vol. VII. No. 9. Page 224. 



