A Treatise on the Theory of Screws

[13, 20 THE THEORY OF SCREWS. ■a the the The conoidal cubic surface represented by this equation has been called the cylindroid*. Each generating line of the surface is conceived to be the residence of a screw, the pitch of which is determined by the expression pa cos2? +pfi sin2Z. When a eylindroid is said to contain a screw, it is not only meant that the screw is one of the generators of the surface, but that the pitch of the screw is identical with the pitch appropriate to the generator with which the screw coincides. We shall first show that it is impossible for more than one cylindroid to contain a given pair of screws 0 and ^>. For suppose that two cylindroids A and B could be so drawn. Then twists about 0 and </> will compound into a twist on the cylindroid A and also on the cylindroid B ('§ 14). There- fore the several screws on A would have to be identical with the screws on B, i.e. the two surfaces could not be different. That one cylindroid can always be drawn through a given pair of screws is proved as follows. Let the two given screws be 0 and <£, the length of their common perpen- dicular be A, and the angle between the two screws be A ; we shall show that by a proper choice of the origin, the axes, and the constants and p^, a cylindroid can be found which contains 0 and If I, m be the angles which two screws on a eylindroid make with the axis of x, and if zlt z2 be the values of z, we have the equations of which the last four are deduced from the first six * This surface has been described by Pliicker (Neue Geometrie des Raumes, 1868-9, p. 97); he arrives at it as follows:—Let 0 = 0, and ß-' = 0 be two linear complexes of the first degree, then all the complexes formed by giving y different values in the expression H + /42' = 0 form a system of which the axes lie on the surface z (x* + y2) - (ft0 - ko)xy = O. The parameter of any complex of which the axis makes an angle w with the axis of x is k = k° cos2 w + k0 sin2 w. Plücker also con- structed a model of this surface. Plücker does not appear to have noticed the mechanical and kinematical properties of the eylindroid which make this surface of so much importance in Dynamics; but it is worthy of remark that the distribution of pitch which is presented by physical considerations is exactly the same as the distribution of parameter upon the generators of the surface, which Plücker fully discussed. The first application of the eylindroid to Dynamics was made by Battaglini, who showed that this surface was the locus of the wrench resulting from the composition of forces of varying ratio on two given straight lines (Sulla serie dei sistemi di forze, Rendie. Ace. di Napoli, 1869, p. 133). See also the Bibliography at the end of this volume. The name eylindroid was suggested by Professor Cayley in 1871 in reply to a request which I made when, in ignorance of the previous work -of both Pliicker and Battaglini, I began to study this surface. The word originated in the following construction, which was then communicated by Professor Cayley. Cut the cylinder x2 + y'2=(p^-pa)'2 in an ellipse by plane z = x, and consider the line æ = 0, y=P^ Pa- If any plane z — e cuts the ellipse in points A, B and the line in C, then CA, CB are two generating lines of the surface.