A Treatise on the Theory of Screws

84 THE THEORY OF SCREWS. [95- a certain screw system of the nth order, be decomposed into n wrenches of intensities a,,... an on n co-reciprocal screws belonging to the same screw system, then the n quantities a1;...an are said to be the co-ordinates of the screw a. Thus the pitch of a will be represented bypxa^ + ...+pna.1f. The virtual coefficient of a and ß will be (p1a1ß1 + ... + pnanßn). We may here remark that in general one screw can be found upon a screw system of the -nth order reciprocal to n — 1 given screws of the same system. For, take 6 — n screws of the reciprocal screw system, then the required screw is reciprocal to 6 — n + n — 1 = 5 known screws, and is therefore determined (§ 25). 96. The Reduced Wrench. A wrench which acts upon a constrained rigid body may in general be replaced by a wrench on a screw belonging to the screw system, which defines the freedom of the body. Take n screws from the screw system of the wth order which defines the freedom, and 6 — n screws from the reciprocal system. Decompose the given wrench into components on these six screws. The component wrenches on the reciprocal system are neutralized by the reactions of the constraints, and may be discarded, while the remainder must compound into a wrench on the given screw system. Whenever a given external wrench is replaced by an equivalent wrench upon a screw of the system which defines the freedom of the body, the latter may be termed, for convenience, the reduced wrench. It will be observed, that although the reduced wrench can be determined from the given wrench, that the converse problem is indeterminate (n< 6). We may state this result in a somewhat different manner. A given wrench can in general be resolved into two wrenches—one on a screw of any given system, and the other on a screw of the reciprocal screw system. The former of these is what we denote by the reduced wrench. This theorem of the reduced wrench ceases to be true in the case when the screw system and the reciprocal screw system have one screw in common. As such a screw must be reciprocal to both systems it follows that all the screws of both systems must be comprised in a single five-system. This is obviously a very special case, but whenever the condition indicated is satisfied it will not be possible to resolve an impulsive wrench into components on the two reciprocal systems, unless it should also happen that the impulsive wrench itself belongs to the five-system*. * I am indebted to Mr Alex. McAulay for having pointed out in his book on Octonions, p. 251, that I had overlooked this exception when enunciating the Theorem of the reduced wrench in the Theory of Screws (1876).