354] THE THEORY OF SCREW-CHAINS. 389 



Let a and /3 be two screw-chains, each consisting of ^ screws, appro 

 priated one by one to the p elements of the mass-chain. If the system 

 receive a twist about the screw-chain /3, while a wrench acts on the screw- 

 chain a, some work will usually be lost or gained ; if, however, no work be 

 lost or gained, then the same will be true of a twist around a acting on 

 a wrench on /3. In this case the screw-chains are said to be reciprocal. The 

 relation may be expressed somewhat differently, as follows : 



If a mass-chain, only free to twist about the screw-chain a, be in equilibrium, 

 notwithstanding the presence of a wrench on the screw-chain j3, then, conversely, 

 a mass-chain only free to twist about the screw-chain ft will be in equilibrium, 

 notwithstanding the presence of a wrench on the screw-chain a. 



This remarkable property of two screw-chains is very readily proved 

 from the property of two reciprocal screws, of which property, indeed, it 

 is only an extension. 



Let ! . . . a M be the screws of one screw-chain, and /3] . . . yS^ those of the 

 other. Let a^, a. 2 , ... a M denote amplitudes of twists on a 1} 2 , &c., and let 

 a i&quot;&amp;gt; 82&quot;, &c., denote the intensities of wrenches on a 1} a 2 , &c. Then, from 

 the nature of the screw-chain, we must have 



/ : / = &amp;lt; : a 2 &quot;=a3 : a 8 &quot;, &c., 

 A : A&quot; = & :&&quot; = & :&&quot;, &c.; 



for as twists and wrenches are compounded by the same rules, the inter 

 mediate screws of the chain require that the ratio of two consecutive 

 amplitudes of the twists about the chain shall coincide with the ratio of the 

 intensities of the two corresponding wrenches. Denoting the virtual 

 coefficient of c^ and j3i by the symbol Br ai /3 lJ we have for the work done by 

 a twist about a, against the screw-chain /3, 



&c., 



while for the work done by a twist about ft against the screw-chain a we 



have the expression 



2a 1 // A / OTaiPl + 2a 2 ^&amp;gt;a^ 2) &c. 



If the first of these expressions vanishes, then the second will vanish also. 



It will now be obvious that a great part of the Theory of Screws may be 

 applied to the more general conceptions of screw-chains. The following 

 theorem can be proved by the same argument used in the case when only a 

 single pair of screws are involved. 



If a screw-chain 6 be reciprocal to two screw-chains a and /3, then 6 will 

 be reciprocal to every screw-chain of the system obtained by compounding 

 twists on a. and /3. 



