372 THE THEORY OF SCREWS. [343 



Let &!, 6.,, :j , 4 be the angles of four screws on the first cylindroid, then the 

 anharmonic ratio will be 



sin (0 4 -6&amp;gt; 3 ) sin (02-00 



From the relation just given between tan0 1( tau^, which applies of course 

 to the other corresponding pair, it will be easily seen that this anharmonic 

 ratio is unaltered when the angles &amp;lt; 1; &amp;lt; 2 , &c., are substituted for 1? 2 , &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 Sj has been chosen on the first 

 cylindroid, we can step from one cylindroid to the next, merely guided in 

 choosing S 2 , S 3 , &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 S lt S 2 , & 3 , &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, ft, 7 

 the screws intermediate to ! and 2 , to /3j and (3. 2 , to y 1 and j 2 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, /3, 7, which neutralize, and that consequently 

 the three twists on the screws a 1 , j3 1} &amp;lt;y l will neutralize, as will also those 

 on a 2 , /3 2 , 72- These six twists must neutralize when compounded in any 

 way whatever. We shall accordingly compound a x and a 2 into one twist 

 on their intermediate screw, and similarly for /^ and /3 2 , and for y l and 73. 

 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 Sj be given, then it is plain that the intermediate screw between Sj 

 and 8 2 , as well as all the other screws of the chain and their intermediate 



