A Treatise on the Theory of Screws

161] THE GEOMETRY OF THE CYLIN DROID. 155 In Fig. 36 will be found a drawing of this curve. The following are the values of the constants adopted : a = 25°; /3 = 26u; A = 18; m = 28'9; with which the equations become x = '9y tan (0 — 25°), y = 20 — 66 sin 20. The curve was plotted down on “papier millimétrique,” and has been copied in reduced size in the figure. The constants were selected after several trials, in order to give a curve that should be at once characteristic, and of manageable dimensions. The distribution of pitch upon the screws of the cylindroid is of fundamental importance in the theory, so that we must express the pitches appropriate to the several points on the cubic. Let p denote the pitch ; then, from the known property of the cylindroid, P=2,o + wicos20, where is a constant. Transforming this result into the co-ordinates of the point on the cubic, we have (a;2 — y2 cos2 ß) cos 2a + 2xy cos ß sin 2a p — Po m yi cog2 ß 161. Chord joining Two Screws of Equal Pitch. As the pitches of the two screws, defined by + 6 and — 6, are equal, the chord in question is found by drawing the line through the points x', y' and x", y", respectively, where x — y' cos ß tan (0 — a), y' =h sec ß — m cosec ß sin 20, x" — y" cos ß tan (— 6 — a), y" = h sec ß + m cosec ß sin 26. After a few reductions, the required equation is found to be xm (cos 26 + cos 2a) + y (h sin ß + m cos ß sin 2a) — h2 tan ß + m2 cot ß sin2 20 = 0. If this chord passes through the origin, then — h2 tan2 ß + m2 sin2 26 — 0 ; or, h tan ß + m sin 26 = 0. But this is obviously necessary; for from the geometry of the cylindroid it is plain that 6 must then fulfil the required condition.