343] THE THEORY OF SCREW-CHATNS. 371 



This important proposition can be easily demonstrated by the aid of 

 fundamental principles. 



The two first chains, a and /3, 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, 8 l} is chosen arbitrarily on the first cylindroid, then 

 8 2 , 6 3 , &c., ... 8^, are all determined uniquely; for a twist about ^ can be 

 decomposed into twists about x and &. The amplitudes of the twists on 

 ofj and & determine the amplitudes on or 2 and /3 2 by the property of the 

 intermediate screws which go to make up the screw-chains, and by com 

 pounding the twists on a., and /8 2 we obtain S 2 . If any other screw of the 

 series, for example, 8 2 , had been given, then it is easy to see that ^ and 

 all the rest, 8 3 , ... B^, 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 l , fa be the angles for two corresponding screws 

 on the first arid second cylindroids, then we must have some relation which 

 connects tan 1 and tan^j. But this relation is to be consistent with the 

 condition that in every case one value of tan l is to correspond to one 

 value of tan fa, and one value of tan fa to one value of tan lf 



If for brevity we denote tan 6 l 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 =U, 



where U is some function of x and where U 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) 



/ / / 



px +q 



The relation between tan 0^ and tan fa will therefore have the form which 

 may generally be thus expressed, 



a tan 1 tan fa+b tan l + c tan &amp;lt;f&amp;gt; 1 + d = 0. 



242 



