A Treatise on the Theory of Screws

343] THE THEORY OF SCREW-CHAINS. 371 This important proposition can be easily demonstrated by the aid of fundamental principles. The two first chains, a and ß, will be sufficient to determine the entire series of cylindroids. When the third chain, 7, is also given, the construction of additional chains can proceed by the anharmonic equality without any further reference to the ratios of the amplitudes. When any screw, 8j, is chosen arbitrarily on the first cylindroid, then 82, 831 &c., ... are all determined uniquely; for a twist about can be decomposed into twists about a, and ß^. The amplitudes of the twists on ctj and ß-i determine the amplitudes on a2 and ßs by the property of the intermediate screws which go to make up the screw-chains, and by com- pounding the twists on a2 and /32 we obtain 82. If any other screw of the series, for example, X,, had been given, then it is easy to see that 8, and all the rest, 83,... 8^, are likewise determined. Thus for the two first cylindroids, we see that to any one screw on either corresponds one screw on the other. If one screw moves over the first cylindroid then its correspondent will move over the second and it will now be shown that these two screws trace out two homographic systems. Let us suppose that each screw is specified by the tangent of the angle which it makes with one of the principal screws of its cylindroid. Let 0lt fa be the angles for two corresponding screws on the first and second cylindroids, then we must have some relation which connects tan and tan^. But this relation is to be consistent with the condition that in every case one value of tan 0, is to correspond to one value of tan<£x, and one value of tan fa to one value of tan öj. If for brevity we denote tan 01 by x and tan fa by x then the geometrical conditions of the system will give a certain relation between x and x. The one-to-one condition requires that this relation must be capable of being expressed in either of the forms x=U'x' — Ü, where U' is some function of x' and where Ü is a function of x. From the nature of the problem it is easily seen that these functions are algebraical and as they must be one valued they must be rational. If we solve the first of these equations for x the result that we obtain cannot be different from the second equation. The first equation must therefore contain x' only in the first degree in the form (see Appendix, Note 7) The relation between tan 0, and tan fa will therefore have the form which may generally be thus expressed, a tan (9, tan fa + b tan + c tan fa + d = 0. 24—2