A Treatise on the Theory of Screws

 372 THE THEORY OF SCREWS. [343 Let 02, 02, 0t be the angles of four screws on the first cylindroid, then the anharmonic ratio will be sin (04 — 02) sin (03 — 0ß sin (04 — 03) sin (02 — 0ß ' From the relation just given between tan 0lt tan </>,, which applies of course to the other corresponding pair, it will be easily seen that this anharmonic ratio is unaltered when the angles cj>2, &c., are substituted for 0lt 02, &c. We have, therefore, shown that the anharmonic ratio of four screws on the first cylindroid is equal to that of the four corresponding screws on the second cylindroid, and so on to the last of the cylindroids. As soon, therefore, as any arbitrary screw has been chosen on the first cylindroid, we can step from one cylindroid to the next, merely guided in choosing 32, 83, &c., by giving a constant value to the anharmonic ratio of the screw chosen and the three other collateral screws on the same cylindroid. Any number of screw-chains belonging to the system may be thus readily constructed. This process, however, does not indicate the amplitudes of the twists appropriate to 8,, 82, &c. One of these amplitudes may no doubt be chosen arbitrarily, but the rest must be all then determined from the geometrical relations. We proceed to show how the relative values of these amplitudes may be clearly exhibited. The first theorem to be proved is that in the three screw-chains a, ß, y the screws intermediate to a2 and a.,, to ßT and ß2, to y, and y2 are co- cylindroidal. This important step in the theory of screw-chains can be easily inferred from the fundamental property that three twists can be given on the screw-chains a, ß, y, which neutralize, and that consequently the three twists on the screws alt ß1} y3 will neutralize, as will also those on a2, ß2, y2. These six twists must neutralize when compounded in any way whatever. We shall accordingly compound and a2 into one twist on their intermediate screw, and similarly for ß2 and ß2, and for and y2. We hence see that the three twists about the three intermediate screws must neutralize, and consequently the three intermediate screws must be co-cylindroidal. We thus learn that in addition to the several cylindroids containing the primary screws of each of the system of screw-chains about which a mass- chain with two degrees of freedom can twist, there are also a series of secondary cylindroids, on which will lie the several intermediate screws of the system of screw-chains. If 8, be given, then it is plain that the intermediate screw between S, and 82, as well as all the other screws of the chain and their intermediate