A Treatise on the Theory of Screws

[348, 382 THE THEORY OF SCREWS. It thus appears that four straight lines in one system may be chosen arbitrarily to correspond respectively with four straight lines in the other system; and that one force being chosen on one of these straight lines in one system, the corresponding force may be chosen arbitrarily on the corre- sponding straight line in the other system. This having been done, the relation between the two systems is completely defined. From the case of parallel projections in two planes it is easy to pass to the case which will serve our present purpose. Instead of the straight lines in the two planes we shall take screws in two »-systems. Instead of the forces on the lines we may take either twists or wrenches on the screws. More generally it will be better to use Plucker’s word “ Dyname,” which we have previously had occasion to employ (§ 260) in the sense either of a twist or a wrench, or even a twist velocity. We shall thus have a Dyname in one system corresponding to a Dyname in the other. Let us suppose that a Dyname on a screw of one n-system corresponds uniquely to a different Dyname on a screw of another n-system. The two n-systems may be coincident but we shall treat of the general case. In the first place it can be shown that if any number of Dynames in the first system neutralize, their corresponding Dynames in the second system must also neutralize. Take n screws of reference in one system, and also n screws of reference in the corresponding system. Let 0 be the Dyname in one system which corresponds to </> in the other; 0 can be completely resolved into component Dynames of intensities ... 0n on the n screws of reference in the first system and in like manner can be resolved into n components of intensities <f>1, ... <fjn on the screws of reference in the second system (n = < 6). From the fact that the relation between 0 and <[> is of the one-to-one type the several components are derived from 0lt... 0n by n equations which may be written $1=(ll)01 + (12)02...+(ln)0n, ij>n = (ml) 0} + (n2) 02... + (nn) 0n, in which (11), (12), &c. must be independent of both 0 and <j>, for otherwise the correspondence would not be unique. If there be a number of Dynames in the first system the sums of the intensities of their components on the n screws of reference may be expressed as ... 30,j respectively. In like manner the sums of the intensities of the components of their correspondents on the screws of reference of the second system may be represented by ,... respectively. We therefore