A Treatise on the Theory of Screws

240 THE THEORY OF SCREWS. [226, screw of the enclosing (w + l)-system whose co-ordinates are proportional to 1 dlf, 1 Pi dr/i Pn+1 This we shall term the polar screw of y with respect to the quadratic n-system. It is supposed, of course, that the screws of reference are co- reciprocals. If a and ß be two screws of an enclosing (n + l)-system, and if y and £ be their respective polar screws with reference to a quadratic n-system, then when a is reciprocal to £ we shall have ß reciprocal to r/. For we have, where h is a common factor, , 1 dUa , 1 dUa /i?71 — - y , ... IbYln+x — j — , Pi Pn+1 dctn+1 whence h^PiPißi + ... +pn+i Vn+1 ßn-\-i) — bdaß. If therefore ß and y are reciprocal the left-hand member of this equation is zero and so must the right-hand member be zero. But the symmetry shows that £ and a are in this case also reciprocal. We may in such a case regard a and ß as two conjugate screws of the quadratic n-system. As a first illustration of the relation between a screw and its polar, we shall take for Ua = 0, the form + p2a22 + ... + p6al - p (a,2 + a22 + ... + a62 + 2aja2 cos (12) ... ) = 0. This means of course that Ua = 0 denotes every screw which has the pitch p. Take any screw a and draw a cylindroid through a. The two screws of pitch p on this cylindroid belong to U and a fourth screw 6 may be taken on this cylindroid so that a, 9, and the two screws of pitch p form an harmonic pencil. By drawing another cylindroid through a another screw of the ö-system can be similarly constructed. If these five cylindroids be drawn through a we can construct five different screws of the 0-system. To these one screw will be reciprocal, and this is the polar of a. We have thus the means of constructing the polar of a. Seeing however that Ua = 0 includes nothing more or less than all the 2? pitch screws in the universe and that in the construction just given for the polar of a there has been no reference to the screws of reference, sym- metry requires that the polar of a must be a screw which though different from a must be symmetrically placed with reference thereto. The only method of securing this is for the polar of a with respect to this particular function to lie on the same straight line as a.