A Treatise on the Theory of Screws

228J FREEDOM OF THE FOURTH ORDER. 243 of the screws in the third degree. We can express this condition as a determinant by employing a canonical system of co-reciprocals. For if two screws 0 and <f> intersect, then there must be some point x, y, z which shall satisfy the six equations (§ 43): (a6 + a6) y - («3 + a4) z = a - a2) -pa (ax + a„), (aj + as) z - (a5 + a6) x = b (a3 - a4) - p« (a3 + a4), (as + a4) x - (otj + a2) y = c (a5 - a6) -p* (a5 + a„), (65 + 0„) y - (03 + 04) z = a (di - d2) -pe (di + 02), (di + 02) z - (0B + 08) x = b (ds - d^ - pe (d3 + di), (03 + d^x - (di + d2) y = c (ds - d6) -pe (d, + dt). From these equations we eliminate the five quantities x, y, z, pe, p„ and the required condition that d and a shall intersect, is given by the equa- tion o (a6 + «s)> - (a3 + a4), (aj + a2), 0 , a (aj — a2) - (a6 + a6), 0 (aj + a2), (a3 + a4), o , b (a3 - a4) (as + a4), - («1 + a2), o (a5 + a,), o , c (as - a6) 0 (d3 + d6), ~(d3+di), 0 (di + d2), a (di - d2) — (d5 + d6). o (di + d2), o (d3 + di), b(d3-di) (d3+di), -(di + d2), o o (ds+ds), c (ds - ds) Four homogeneous equations between the co-ordinates of d indicate that the corresponding screw lies on a certain ruled surface. Let us suppose that the degrees of these equations are I, m, n, r respectively, then the degree of the ruled surface must not exceed Slmnr. For express the condition that d shall also intersect some given screw a, we then obtain a fifth homogeneous equation containing the co-ordinates of 6 in the third degree. The determination of the ratios of the six co-ordi- nates di, ... d3 is thus effected by five equations of the several degrees I, m, n, r, 3. For each ratio we obtain a system of values equal in number to the product of the degrees of the equations, i.e. to 3lmnr. This is accord- ingly a major limit to the number of points in which in general a pierces the surface, that is to say, it is a major limit to the degree of the surface. Of course we might affirm that it was the degree of the surface save for the possibility that through one or more of the points in which a met the surface more than a single generator might pass. As an example, we may take the simple case of the cylindroid, in which I, m, n, r being each unity the locus is of the third degree. The screws of 16—2