A Treatise on the Theory of Screws

[393-397 432 THE THEORY OF SCREWS. Freedom we obtain a cubic for X. The three roots of this cubic correspond to three g screws. Take the 3 corresponding to one of the g screws, then, of course, 3 will not, in general, belong to the four-system. We can, however, assign to 3 any pitch we like, and as it intersects g at right angles, it must cut two other screws of equal pitch on the cylindroid (§22). Give to 3 a pitch equal and opposite to that of the two latter screws, then 3 is reciprocal to the cylindroid, and therefore it belongs to the four-system. We thus have a permanent screw of the system, and accordingly we obtain the following result:— In the case of a rigid body with freedom of the fourth order there are, in general, three, and only three, permanent screws. 394. Freedom of the Fifth and Sixth Orders. When a rigid body has freedom of the fifth order, the screws about which the body can be twisted are all reciprocal to a single screw p. In general, p does not lie on the system prescribed by the equation which the co-ordinates of all possible g screws have to satisfy. It is therefore, in general, not possible that the reaction of the constraints can provide an g. There are, however, three screws in any five-system which possess the property of permanent screws without however making any demand on the reaction of the constraints. The existence of these screws is thus demonstrated:— Through the centre of inertia of the body draw the three principal axes, then, on each of these axes one screw can always be found which is reciprocal to p. Each of these will belong to the five-system, and it is obvious from the property of the principal axes, that if the body be set twisting about one of these screws it will have no tendency to depart therefrom. A body which has freedom of the sixth order is perfectly free. Any screw on one of the principal axes through the centre of inertia is a permanent screw, and, consequently, there is in this case a triply infinite number of permanent screws. I. II. III. IV. V. VI. 395. Summary. The results obtained show that for a rigid body with the several degrees of freedom the permanent screws are as follows :— No. of Permanent Screws _____ 1 _____ 1 _____ 3 _____ 3 _____ 3 Triply infinite