162] THE GEOMETRY OF THE CYLINDROID. 159 



the cone breaks up into two planes. The nature of these planes is easily 

 seen. One of them, A, must be the plane perpendicular to the generator 

 through ; the other, B, is the plane containing 0, and the screw of equal 

 pitch to that of the screw through 0. These planes intersect in a ray, L, 

 and it must first be shown that L is a tangent to the cyliridroid. 



Any ray intersecting one screw on a cylindroid at right angles must cut 

 the surface again in two screws of equal pitch ; consequently L can only 

 meet the surface in two distinct points, each of which has the pitch of the 

 generator through 0. It follows that L must intersect the surface at two 

 coincident points 0, i.e. that it is a tangent to the cylindroid at 0. 



Let any plane of section be drawn through 0. This plane will, in 

 general, intersect A and B in two distinct rays : these are the two screws 

 reciprocal to the cylindroid, and they are accordingly the two tangents from 

 to the parabola we have been discussing. The only case in which these 

 two rays could coalesce would occur when the plane of section was drawn 

 through L ; but the two tangents to a parabola from a point only coalesce 

 when that point lies on the parabola. At a point where the parabola meets 

 the cubic, L must needs be a tangent both to the parabola and to the cubic, 

 which can only be the case if the two curves are touching. We have thus 

 proved that the parabola must have triple contact with the cubic. 



There are thus three points on the cubic which have the property that 

 the tangent intersects the curve again in a point of equal pitch to that of the 

 point of contact. We thus learn that all the screws of a four-system which 

 lie in a plane touch a parabola having triple contact with the reciprocal 

 cylindroid. 



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 two screws whose pitch is equal 

 to that of P. The other 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 Fig. 36. 



From the equation of the cylindroid, 



z (a? + 2/ 2 ) = 2ma?y, 



it follows that the plane at infinity cuts the surface in three straight lines 

 on the planes, 



* = 0, 



x iy = 0. 



