158 THE THEORY OF SCREWS. [162 



The latus rectum is 



4 cos /3 (h + m cot /3 sin 2a). 



The values of the two equal pitches (p) on the pair of screws are thus 

 expressed in terms of the abscissa x of the point in which the chord touches 

 the envelope by means of the equation 



From any point P, on the cubic, two tangents can be drawn to the 

 parabola. Each of these tangents must intersect the cubic in a pair of screws 

 of equal pitch. One tangent will contain the other screw whose pitch is equal 

 to that of P. The second tangent passes through two screws of equal pitch 

 in the two other points which, with P, make up the three intersections with 

 the cubic. As the principal screws of the cylindroid are those of maximum 

 and minimum pitch respectively, it follows that the tangents at these points 

 will also touch the parabola. These common tangents are shown in the 

 figure. 



This parabola is drawn to scale in Fig. 36. The equation employed was 



2 



5 + 



When the figure was complete, it was obvious that the parabola touched 

 the cubic, and thus the following theorem was suggested : 



The parabola, which is the envelope of chords joining screws of equal pitch, 

 touches the cubic in three points. 



The demonstration is as follows : To seek the intersections of the 

 parabola with the cubic, we substitute, in the equation of the parabola, the 

 values 



x = h tan (6 a) m cot /3 sin 20 tan (9 a), 



y = h sec /3 m cosec /3 sin 20, 



This would, in general, give an equation of the sixth degree for tan 9. It 

 will, however, be found in this case that the expression reduces to a perfect 

 square. The six points in which the parabola meets the cubic must thus 

 coalesce into three, of which two are imaginary. The values of 9 for these 

 three points are given by the equation 



h tan (9 - a) - m cot (sin 26 tan (9 - a) + 2 cos 20} = 0. 



We can also prove geometrically that the parabola touches the cubic at 

 three points. 



In general, a cone of screws reciprocal to the cylindroid can be drawn 

 from any external point. If the point happen to lie on the cylindroid, 



