A Treatise on the Theory of Screws

268] EMANANTS AND PITCH INVARIANTS. 283 It is easily seen that this equation must reduce to the form A = 0. In fact, seeing it expresses the solution of the problem of finding a screw of maximum pitch, and that the choice may be made from a system of the sixth order, that is to say, from all conceivable screws in the universe it is obvious that the equation could assume no other form. What we now propose to study is the manner in which the necessary evanescence of the several' coefficients is provided for. After the equation has been expanded we shall suppose that each term is divided by the coefficient of xs that is, by pipiPaptPsP«- From any point draw a pencil of rays parallel to the six screws. On four of these rays, 1, 2, 3, 4, we can assign four forces which equilibrate at the point. Let these magnitudes be Xlt X2, X3, X4. We can express the necessary relations by resolving these four forces along each of the four directions successively. Hence X, + X2 cos (12) + X3 cos (13) + X4 cos (14) = 0. cos (21) + X2 + X3 cos (23) + Xt cos (24) = 0. X cos (31) + X2 cos (32) + X3 + X cos (34) = 0. X, cos (41) + X2 cos (42) + X3 cos (43) + XA = 0. Eliminating the four forces we have 1, cos (12), cos (13), cos(14) cos (21), 1, cos (23), cos(24) cos (31), cos (32), 1, cos(34) cos (41), cos (42), cos (43), 1 Thus we learn that every determinant of this type vanishes identically. Had we taken five or six forces at the point it would, of course, have been possible in an infinite number of ways to have adjusted five or six forces to equilibrate. Hence it follows that the determinants analogous to that just written, but with five and six rows of elements respectively, are all zero. These theorems simplify our expansion of the original harmonic deter- minant. In fact, it is plain that the coefficients of x2, of x, and of the absolute term vanish identically. The terms which remain are as follows:— xa + Ax6 + Bx2 + Cxä = 0.