A Treatise on the Theory of Screws

164 THE THEORY OF SCREWS. [165 The envelope of this chord is found to be 0 = +a'2 ||rø.asin22a, + y2 I |m2 cos2 ß cos2 2a — hm sin ß cos ß sin 2a — h2 sin2 ß 4- X cos 2a cos2 ß + ^m2 X" cos2 ß, + xy I — 9 sin 2a cos 2a cos ß — mh cos 2a sin ß - % m2X sin 2a cos ß - mhX sin ß, + x I + mb2cos 2a tan ß + hm2X sin 2a + mh? Xtan ß, + y I - m?h cos ß + mh? sin ß sin 2a + 2/i3 sin ß tan ß - m?hX cos ß cos 2a, + I m?h? — h4 tan2 ß. Using the data already assumed in § 160, and, with the addition now made of taking X to be — J, the equation reduces to 122a;2 — 203?/2- 161«y- 2417«- 5003y+ 245436 = 0; which, for convenience of calculation, I change into x = 1’6 + 21’6 sec </> + 47’5 tan </>, y = — 12'8 + 32’8 sec </>. This hyperbola has been plotted down in Fig. 36. It obviously touches the cubic at three points. I had not anticipated this until the curves were care- fully drawn; but, when the theorem was suggested in this manner, it was easy to provide the following demonstration:— The cone of reciprocal chords drawn through any point P breaks up into a pair of planes when P lies on the cylindroid. (I use the expression, reciprocal chord, to signify the transversal drawn across a pair of reciprocal screws on the cylindroid. This is very different from a screw reciprocal to the cylindroid.) For, take the screw reciprocal to that which passes through P. Then the plane X, through P and this screw, is obviously one part of the locus. Draw through P any transversal across a pair of reci- procals on the cylindroid, then the plane Y, through the centre and this transversal, will be the other part of the locus, this pair of planes, X and Y, intersect in a ray which we shall call & A plane of section through P will, of course, usually cut the two planes in two rays, and these will be the two reciprocal chords through P. But suppose the plane of section happened to pass through S, then there will be only one reciprocal chord through P, and this will, of course, be >S’. Now, S must be a tangent to the cylindroid at P. Every chord through P, in the plane of Y, must cut the surface again in a pair of reciprocal points. To this 8 must be no exception, and as it lies in X, it intersects the screw