A Treatise on the Theory of Screws

206 THE THEORY OF SCREWS. [205- The angle between two screws is equal to I, i times the logarithm of the (in- harmonic ratio in which their corresponding chord is divided by the infinite pitch conic. The reader will be here reminded of the geometry of non-Euclidian space, in which a magnitude, which in Chapter XXVI. is. called the Intervene, analogous to the distance between two points, is equal to §i times the logarithm of the anharmonic ratio in which their chord is divided by the absolute. We have only to call the conic of infinite pitch the absolute, and the angle between two screws is the intervene between their corresponding points. 206 Screws at Right Angles. If two screws, 0 ami be at right angles, then dfi>! + 0fi>2+0fi>3 = O. In other words, 0 and </> are conjugate points of the conic of infinite pitch, d^+df+d^o. All the screws at right angles to a given screw lie on the polar of the point with regard to the conic of infinite pitch. Hence we see that all the screws perpendicular to a given screw lie on a cylindroid. This is otherwise obvious, for a screw can always be found with an axis parallel to a given direction. If, therefore, a cylindroid of the system be taken, a screw of the system parallel to the nodal axis of that cylindroid can also be found, and thus we have the cylindroid and the screw, which stand in the relation of the pole and the polar to the conic of infinite pitch. A point on the conic of infinite pitch must represent a screw at right angles to itself. Every straight line cuts the conic of infinite pitch in two points, and thus every cylindroid has two screws of infinite pitch, and each of these screws is at right angles to itself. In general, the direction cosines of the nodal axis of a cylindroid are proportional to the co-ordinates of the pole of the line corresponding to the cylindroid with respect to the conic of infinite pitch. 207. Reciprocal Screws. If dlt d2, d3 be the co-ordinates of a screw, and </>,, <f>2, </>3 those of another screw, then it is known, § 37, that the condition for these two screws to be reciprocal is Pidfih + p-fifi* + p3dfi>s = 0. We are thus led to the following theorem, which is of fundamental importance in the present investigation