A Treatise on the Theory of Screws

166 THE THEORY OF SCREWS. [165- There are two critical cases in which this expression vanishes. It does so if h tan ß + m sin 2a = 0; where the plane of section is tangential to the cylindroid. But we also note that the discriminant will vanish if h = 0, i .e. if the plane of section passes through the centre of the cylindroid. We might have foreseen this from the results of the last article; for the plane of Y is a central section, and the hyperbola has evidently degenerated, for all the reciprocal chords, instead of touching an hyperbola, merely pass through the common apex P. The case of the central section is therefore of special interest. 166. The Central Section of the Cylindroid. By this we mean a section of the surface, special in no other sense, save that it passes through the centre of the surface. The equation to the central section is (§ 160) if sin ß cos2ß + yx1 sin ß - mir? sin 2a + 2mxy cos ß cos 2a + my1 sin 2a cos2/3 = 0. The chord joining points of equal pitch + 0 is x (cos 20 + cos 2a) + y cos ß sin 2a + m cot ß sin2 20 — 0. The apex P through which all reciprocal chords pass is m (X2 — 1) cot ß (X + cos 2a) X= 1 + 2X cos 2a + X2 ’ m (X2 — 1) cosec ß. sin 2a __ y~ 1 + 2Å cos 2a + X2 and in general the co-ordinates of a point on the cubic are x — y cos ß tan (0 — a), y = — m cosec ß sin 20. One of these curves may be conveniently drawn to scale, from the equations x = '9y tan (0 — 25), y = — 66 sin 20. The parabola, which is the envelope of equal pitch-chords, would in this case have as its equation 2Z = _51-(6+g)