164] THE GEOMETRY OF THE CYLINDROID. 161 



of the cubic. If the coefficient of a? become zero, the required decomposition 

 takes place, for y is then a factor. The necessary and sufficient condition 

 for the plane of section being a tangent, is therefore 



ra sin 2a + h tan /3 = 0. 



When this is the case, the three expressions of A, B, C may be divided by 

 a common factor, 



2m cos (6 a) cos (6&quot; a). 

 and we have 



A = cos (& + 0&quot;), 



B = - cos /3 sin (& + 0&quot;), 



G = - 2m cot 13 sin (a + ) sin (a + 0&quot;). 



If the screws be of equal pitch, 9 + 0&quot; = 0, the coefficient of y disappears, 

 and we see that all the chords are merely lines parallel to the axis of y, which 

 is parallel to one of the axes of the ellipse. 



The equation to the chord then becomes 



(cos 2a + cos 20) [x + m cot ft (cos 2a cos 20)} = 0. 



For a given value of x there are two values of corresponding to the two 

 chords that can be drawn through the point. One of these chords is parallel 

 to y, and has a obtained from the equation 



cc + m cot /3 (cos 2a cos 20) = 0. 



7T 



The other value of is -- a, from the equation 



2 



cos 2a+ cos 20 = 0. 



This is independent of x, as might have been foreseen from the fact that 

 the two screws of equal pitch are in this case the line in the section and the 

 other screw of equal pitch. The latter cuts the section in a certain point, and, 

 of course, all chords through this point meet the curve in two screws of equal 

 pitch. 



164. Reciprocal Screws. 



Another branch of the subject must now be considered. We shall first 

 investigate the following general problem : 



From any point, P, a series of transversals is drawn across each pair 

 of reciprocal screws on the cyliiidroid. It is required to determine the cone 

 which is the locus of these transversals. We shall show that this is a cone 

 of the second degree. 



B. 11 



