A Treatise on the Theory of Screws

150 TI1E THEORY OK SCREWS. [159 from the same principles. If a and ß are screws of zero pitch, then any reciprocal transversal 0 will be also of zero pitch; and as p must be reciprocal to 0, it will follow that the pitch of p must be equal to the product of the shortest perpendicular distance between p and 0, and the tangent of the angle between the two lines. In short, the pitch of p must simply be equal to what is sometimes called the moment between p and 0. We are also led to the following construction for the cylindroid. Draw a plane pencil of rays and another ray L, situated anywhere. Then the common perpendiculars to L and the several rays of the pencil trace out the cylindroid. I have already mentioned (p. 20) that the first model of the cylindroid was made by Pliicker in illustration of his Neue Geometrie des Raumes. The model of the surface which is represented in the Frontispiece was made from my design by Sir Howard Grubb, the cost being defrayed by a grant from the Scientific Fund of the Royal Irish Academy. A hollow cylinder was mounted on a dividing engine and holes were drilled at the calculated points. Silver wires were then stretched across in the positions of the generators and a beautiful model is the result. The equation to the tangent cone drawn from the point x', y', z' to the surface, z (a? + ?/2) — %mxy = 0, is of the fourth degree and is given by equating to zero the discriminant of the following function in &>, w3 (xz — zx') - to2 {yz'~zy + 2m(æ — #')} + w \xz' — zx + 2 m(y — yf \ + zy' — yz'. This cone has three cuspidal edges, and accordingly the model exhibits in every aspect a remarkable tricuspid arrangement. I here give the details of the construction of the much simpler model of the cylindroid figured in Plate II.* A boxwood cylindei-, 0™"15 long and O,,l'O5 in diameter, is chucked to the mandril of a lathe furnished with a dividing plate. A drill is mounted on the slide rest, and driven by overhead gear. The parameter pa— pg (in the present case 0m066) is divided into one hundred parts. By the screw, which moves the slide rest parallel to the bed of the lathe, the drill can be moved to any number z of these parts from its original position at the centre of the length of the cylinder. Four holes are to be drilled for each value of z. These consist of two pairs of diametrically opposite holes. The directions of the holes intersect the * See Transactions of Royal Irish Academy, Vol. xxv. p. 216 (1871); and also Phil. Mag. Vol. xlii. p. 181 (1871); also B. A. Report, Edinburgh, 1871.