A Treatise on the Theory of Screws

488 THE THEORY OF SCREWS. which possess the required property. Choose any screw a and then take any screw ß whose co-ordinates satisfy these two conditions. We shall also use the plane representation of the 3-system. Let I] = 0 be the pitch conic. V = 0 be the imaginary ellipse obtained by equating to zero the expression for the Kinetic Energy. Then the vertices of a common conjugate triangle are of course the principal Screws of Inertia and generally there is only one such triangle. It may however happen that f7 and V have more than a single common conjugate triangle, for let the cartesian co-ordinates of the four intersections of U and V be represented by xi, z/ij æ2, y2; «s, y3; æ4, y«.- As all the points on P" are imaginary at least one co-ordinate of each intersection is imaginary. Suppose yx to be imaginary then it must be conjugate to If therefore the conic U touches V y1 and y3 must be respectively equal to y2 and y . Hence we have only two values of y, and these are conjugate. Substituting these in U and V we see that there can only be two values of æ, and consequently the intersections reduce to two pairs of coincident points. Hence we see that F cannot touch U unless the two conics have double contact. In this case the chord of contact possesses the property that each point on it is a principal Screw of Inertia while the pole of the chord with respect to either conic is also a principal Screw of Inertia. If U and V coincided then every screw of the 3-system would be a principal Screw of Inertia. The general theory on the subject is as follows. Let (7= 0 be the quadratic relation among the co-ordinates of an ra-system which expresses that its pitch is zero. Let 7 = 0 be the quadratic relation among the co-ordinates of a screw if a body twisting about that screw has zero kinetic energy. The discriminant of S ~ U + W equated to zero gives n real roots for A. These roots substituted in the differential coefficients of S equated to zero give the corresponding principal Screws of Inertia. If however there be two equal roots for Å then for these roots every first minor of the discriminant vanishes. In this case $ can be expi'essed as a function of n — 2 linear quantities. Perhaps the most explicit manner of doing this is as follows. Let and let 8 = a^2 + ai202 + + ... + annGn2, IdS 1 dS 2d6ß -Sn~2don-