A Treatise on the Theory of Screws

CHAPTER III . RECIPROCAL SCREWS. 20. Reciprocal Screws. If a body only free to twist about a screw a be in equilibrium, though acted upon by a wrench on the screw ß, then conversely a body only free to twist about the screw ß will be in equilibrium, though acted upon by a wrench on the screw a. The principle of virtual velocities states, that if the body be in equili- brium the work done in a small displacement against the external forces must be zero. That the virtual coefficient should vanish is the necessary and the sufficient condition, or (§ 10) (pa + pß) cos 0 — d sin 0 = 0. The symmetry shows that precisely the same condition is required whether the body be free to twist about a, while the wrench act on ß, or vice versa. A pair of screws are said to be reciprocal when their virtual co- efficient is zero. 21. Particular Instances. Parallel or intersecting screws are reciprocal when the sum of their pitches is zero. Screws at right angles are reciprocal either when they intersect, or when one of the pitches is infinite. Two screws of infinite pitch are reciprocal, because a couple could not move a body which was only sus- ceptible of translation. A screw whose pitch is zero or infinite is reciprocal to itself*. 22. Screw Reciprocal to Cylindroid. If a screw y be reciprocal to two given screws 0 and then y is reciprocal to every screw on the cylindroid (0, <fr). See also Professor Everett, I1'.U.S., Messenger of Mathematics, New Series (1874), No. 39.