A Treatise on the Theory of Screws

350 the theory of screws. [320- can be no further ambiguity. The correspondent in A to every other screw in P is completely known. To show this it is only necessary to take two pairs from A' and P' and the pair just found. We have then three corre- sponding pairs. It has been shown in § 307 how the correspondence is completely determined in this case. Of course the fact that 8 may be any screw on a cylindroid is connected with the fact that in this case the rigid body had one indeterminate element. For each of the possible rigid bodies 6 would occupy a different position on its cylindroidal locus. 321. Systems of the Fifth Order. It remains to consider the case where two screw systems of the fifth order are given, it being known that one of them P is the locus of the impulsive screws which correspond to the several screws of the other system A regarded as instantaneous screws. Let P be the screw reciprocal to P, and A' the screw reciprocal to A. Then from the theorem of § 319 it follows that an impulsive wrench on A' would make the body commence to move by twisting about P'. We thus know five of the coordinates of the rigid body. There remain four inde- terminate elements. Hence we see that, when the only data are the two systems P and A, there is a fourfold infinity in the choice of the rigid body. There are conse- quently four arbitrary elements in designing the correspondence between the several pairs of screws in the two systems. We may choose any two screws y, %, on P, and assume as their two corre- spondents in A any two arbitrary screws a and ß, provided of course that the three pairs A , B, y, a, and %, ß fulfil the six necessary geometrical conditions (§ 304). Two of these conditions are obviously already satisfied by the circumstance that A' and P' are the reciprocals to the systems A and P. This leaves four conditions to be fulfilled in the choice of a and ß. As each of these belongs to a system of the fifth order there will be four coordinates required for its complete specification. Therefore there will be eight disposable quantities in the choice of a and ß. Four of these will be utilized in making them fulfil the geometrical conditions, so that four others may be arbitrarily selected. When these are chosen we have four coordinates of the rigid body which, added to the five data provided by A' and P', completely define the rigid body. 322. Summary. We may state the results of this discussion in the following manner :— If we are given two systems of the first, or the second, or the third order