388 THE THEORY OF SCREWS. [353, 



353. Freedom of the eighth and higher orders. 



If the freedom be of the eighth order, then it is easily shown that the 

 first screw of any other chain may be taken arbitrarily, and that even the 

 second screw may be chosen arbitrarily from a three-system. Passing on to 

 the twelfth order of freedom, the two first screws of the chain, as well as the 

 amplitudes of their twists, may be chosen quite arbitrarily, and the rest of 

 the chain is fixed. In the thirteenth order of freedom we can take the 

 two first twists arbitrarily, while the third may be chosen anywhere on a 

 cylindroid. It will not now be difficult to trace the progress of the chain 

 to that unrestrained freedom it will enjoy when the mass-chain has G^ 

 degrees of freedom, when it is able to accept any displacement whatever. In 

 the last stage, prior to that of absolute freedom, the system will have its 

 position defined by 6/j, - 1 co-ordinates. A screw-chain can then be chosen 

 which is perfectly arbitrary in every respect, save that one of its screws must 

 be reciprocal to a given screw. 



354. Reciprocal Screw-Chains. 



We have hitherto been engaged with the discussion of the geometrical 

 or kinematical relations of a mass-chain of p elements : we now proceed to 

 the dynamical considerations which arise when the action of forces is 

 considered. 



Each element of the mass-chain may be acted upon by one or more 

 external forces, in addition to the internal forces which arise from the reaction 

 of constraints. This group of forces must constitute a wrench appropriated 

 to the particular element. For each element we thus have a certain wrench, 

 and the entire action of the forces on the mass-chain is to be represented by 

 a series of /j, wrenches. Recalling our definition of a screw-chain, it will be 

 easy to assign a meaning to the expression, wrench on a screw-chain. By this 

 we denote a series of wrenches on the screws of the chain, and the ratio of 

 two consecutive intensities is given by the intermediate screw, as before. 

 We thus have the general statement : 



The action of any system of forces on a mass-chain may be represented 

 by a wrench on a screw-chain. 



Two or more wrenches on screw-chains will compound into one wrench 

 on a screw-chain, and the laws of the composition are exactly the same as 

 for the composition of twists, already discussed. 



Take, for example, any four wrenches on four screw-chains. Each set of 

 four homologous screws will determine a four-system; the resulting wrench- 

 chain will consist of a series of wrenches on these four-systems, each being 

 the &quot;parallel projection&quot; of the other. 



