A Treatise on the Theory of Screws

252] HOMOGRAPHIC SCREW SYSTEMS. 267 screw. We may solve this problem in various ways. One of the simplest will be to write the five invariants 12.3 8 13.48 14.58 15.68 16.78 13.2 8’ 14.38’ 15.48’ 16.58’ 17.68' These can be computed from the given eight screws of one system; hence we have five linear equations to determine the ratios of the coefficients of the required eighth screw of the other system. It would seem that of all the invariants of eight screws, five alone can be independent. These five invariants are attributes of the eight-screw system, in the same way that the anharmonic ratio is an attribute of four collinear points. 251. A Physical Correspondence. The invariants are also easily illustrated by considerations of a me- chanical nature. To a wrench on one screw corresponds a twist on the corresponding screw, and the ratio of the intensities of the wrench and twist is to be independent of those intensities. We may take a particular case to illustrate the argument:—Suppose a free rigid body to be at rest. If that body be acted upon by an impulsive system of forces, those forces will constitute a wrench on a certain screw a. In consequence of these forces the body will commence to move, and its instantaneous motion cannot be different from a twist velocity about some other screw ß. To one screw a will correspond one screw ß, and (since the body is perfectly free) to one screw ß will correspond one screw a. It follows, from the definition of homo- graphy, that as a moves over every screw in space, ß will trace out an homo- graphic system.... From the laws of motion it will follow, that if J1 be the intensity of the impulsive wrench, and if V be the twist velocity which that wrench evokes, then F+V will be independent of F and V, though, of course, it is not independent of the actual position of a and ß. 252. Impulsive and Instantaneous Systems. It is known (§ 230) that when seven wrenches equilibrate (or when seven twist velocities neutralize), the intensity of the wrench (or the twist velocity) on any one screw must be proportional to the sexiant of the six non- corresponding screws. Let Fia, Fx,... F78 be the intensities of seven impulsive wrenches on the screws 1, 2,... 7, which equilibrate, then we must have Fie _ 28 _ _ 78 78’