A Treatise on the Theory of Screws

111] FREEDOM OK THE FIRST ORDER. 103 increased by a certain quantity, and all the pitches of the reciprocal system be diminished by the same quantity, then all the first set of screws thus modified are reciprocal to all the second group as modified. Hence, since a screw-system of the nth order consists of all the screws reciprocal to 6 — n screws, it follows that the modified set must still be a screw system. We shall now apply this principle to prove that all the screws Å of any given pitch k, which can be drawn through A, to be reciprocal to a, lie in a plane. Take a screw y, of pitch pa + k, on the same line as a, then we have just shown that all the screws p, of zero pitch, which can be drawn through the point A, so as to be reciprocal to y, lie in a plane. Since p and y are reciprocal, the screws on the same straight lines as p and y will be reciprocal, provided the sum of their pitches is the pitch of y; therefore, a screw X, of pitch k, on the same straight line as p, will be reciprocal to the screw a, of pitch yja; but all the lines p lie in a plane, therefore all the screws X lie in the same plane. Conversely, given a plane and a pitch k, a point A can be determined in that plane, such that all the screws drawn through A in the plane, and possessing the pitch k, are reciprocal to a. To each pitch ky, k.2>..., will correspond a point A.,...; and it is worthy of remark, that all the points Alt A2 must lie on a right line which intersects a at right angles; for join -41, -42, then a screw on the line A^.j, which has for pitch either k, or k2, must be reciprocal to a; but this is impossible unless ylp42 intersect a at a right angle. 111. Equilibrium. If a body which has freedom of the first order be in equilibrium, then the necessary and sufficient condition is, that the forces which act upon the body shall constitute a wrench on a screw of the screw system of the fifth order, which is reciprocal to the screw which defines the freedom. We thus see that every straight line in space may be the residence of a screw, a wrench on which is consistent with the equilibrium of the body. If two wrenches act upon the body, then the condition of equilibrium is, that, when the two wrenches are compounded by the aid of a cylindroid, the single wrench which replaces them shall lie upon that one screw of the cylindroid, which is reciprocal to a (§ 26). We can express with great facility, by the aid of screw co-ordinates, the condition that wrenches of intensities 0", <f>", on two screws 0, </>, shall equilibrate, when applied to a body only free to twist about a. Adopting any six co-reciprocals as screws of reference, and resolving each of the wrenches on 0 and ø into its six components on the six screws of