A Treatise on the Theory of Screws

________________ ______________________ _________________________________ _______ ______ 209] PLANE REPRESENTATION OF THE THIRD ORDER. 209 By changing the sign of a, and then changing the signs of ß and of y, we shall obtain the four points iu which the pencil 0(Pn P2, Pt, Pß cuts the axis </>3 = 0. If four values of be represented by k, I, m, n, the required <Pa anharmonic ratio is, of course, (n — l)(m — k) (n — m) (I — k) ’ and after a few reductions we find that this becomes But we have et^p^p^ ________________ a^yW^-ßW.^)' _________________ Eliminating fp2, we find that 022 and 0<? disappear also when we make a2 = p2-p3, ß2=ps~Pi, 72 = Pi~P-2, and we obtain the following result: _____ 3 (P^PffP^ = P'-: p' x , Pt-Pt Pi-p which gives the following theorem:— Measure off distances plt p2, p3, p, from an arbitrary point on a straight line, then the anharmonic ratio of the four points thus obtained is equal to the anharmonic ratio subtended by any point of the p-pitch conic at the four points of indeterminate pitch. It is possible without any sacrifice of generality to make the zero-pitch conic a circle. For take three angles A, B, G whose sum is 180° and such that the equations sin 2/1 _ sin 2B _ sin 2G Pi Pt Pt are satisfied where plt p2, p3 are the three principal pitches of the three- system. If the fundamental triangle has A, B, G for its angles then the equation a/ sin 2J + a22 sin 2B + «32 sin 2G = 0, is the equation of the zero-pitch conic. It is however a well-known theorem in conics that this equation represents a circle with its centre at the ortho- centre, that is, the intersection of the perpendiculars from the vertices of the triangle on its opposite sides. We thus have as the system of pitch conics a,2 sin 2 A + a22 sin 2B + «32 sin 2G — p (a2 + a22 + a32) = 0. B. 14