A Treatise on the Theory of Screws

256 THE THEORY OF SCREWS, [236 Let the radius vector of length pe + a = R be marked off along each screw 9 drawn through P, then the equation becomes RX = hZ, or squaring (X2 + F2 + Z2) Xt = A2Z2. This represents a surface of the fourth degree. A model of the surface has been constructed*. It is represented in Fig. 41, and from its resemblance to the valves of a scallop shell the name pectenoid is suggested. The geometrical nature of a pectenoid is thus expressed. Given a screw a of pitch pa and a point 0 situated anywhere. If a screw 9 drawn through 0 be reciprocal to a. then the extremity of a radius vector from 0 along 9 equal to pa + pe will trace out a pectenoid. All pectenoids are similar surfaces, they merely differ in size in accordance with the variations of the quantity h. The perpendicular from 0 upon « is a nodal line, and this is the only straight line on the surface. The pectenoid though unclosed is entirely contained between the pair of parallel planes Z = + h, Z = — h. Sections parallel to the plane of Z are hyperbolas. Any plane through the nodal line cuts the pectenoid in a circle. A straight wire at right angles to the nodal line marked on the model indicates the screw reciprocal to the five-system. A second wire starts from the origin and projects from the surface. It is introduced to show concisely what the pectenoid expresses. If this wire be the axis of a screw 9 whose pitch when added to the pitch of the screw a, is equal to the intercept from the origin to the surface then the two screws are reciprocal. The interpretation of the nodal line is found in the obvious truth that when two screws intersect at right angles they are reciprocal whatever be the sum of their pitches. One of the circular sections made by a plane drawn through the nodal line is also indicated in the model. The physical interpretation is found in the theorem already mentioned, that all screws of the same pitch drawn through the same point and reciprocal to a given screw will lie in a plane. With the help of the pectenoid we can give another proof of the theorem that all the screws of a four-system which can be drawn through a point lie on a cone of the second degree (§ 123). Let 0 be the point and let a and ß be two screws on the cylindroid reciprocal to the system. Let 9 be a screw through 0 belonging to the four-system and therefore reciprocal to a and to ß. Then for the pectenoid relating to 0 and a, we have (pe +Pa) M - kN = 0, where M = 0, N = 0 represent planes passing through 0. * Transactions of the Royal Irish Academy, Vol. xxv. Plate xn. (1871).