272 THE THEORY OF SCREWS. [258, 



258. Corresponding Screws defined by Equations. 



It is easy to state the matter analytically, and for convenience we shall 

 take a three-system, though it will be obvious that the process is quite 

 general. 



Of the six screws of reference, let three screws be chosen on the three- 

 system, then the co-ordinates of any screw on that system will be a 1( 2 , 3 , 

 the other three co-ordinates being equal to zero. The co-ordinates of the 

 corresponding screw ft must be indeterminate, for any screw of a four-system 

 will correspond to ft. This provision is secured by /3 4) /3 5 , /3 f) remaining quite 

 arbitrary, while we have for @ lt j3 2 , @ 3 the definite values, 



If we take /3 4 , /3 5 , /3 6 all zero, then the values of ($ l} /3 2 , /3 3 , just written, give 

 the co-ordinates of the special screw belonging to the three-system, which 

 is among those which correspond to a. 



As a moves over the three-system, so will the other screw of that system 

 which corresponds thereto. There will, however, be three cases in which the 

 two screws coincide ; these are found at once by making 



Pi = p&amp;lt;*i , /3 2 = p* 2 ; /3 3 = pa 3 , 

 whence we obtain a cubic for p. 



It is thus seen that generally n screws can be found on an ?z-system, so 

 that each screw shall coincide with its correspondent. As a dynamical 

 illustration we may give the important theorem, that when a rigid body 

 has n degrees of freedom, then n screws can always be found, about any 

 one of which the body will commence to twist when it receives an impulsive 

 wrench on the same screw. These screws are of course the principal screws 

 of inertia ( 84). 



259. Generalization of Anharmonic Ratio. 



We have already seen the anharmonic equality between four screws on a 

 cylindroid, and the four corresponding screws ; we have also shown a quasi 

 anharmonic equality between any eight screws in space and their cor 

 respondents. More generally, any n + 2 screws of an w-system are connected 

 with their n + 2 correspondents, by relations which are analogous to an 

 harmonic properties. The invariants are not generally so simple as in the 

 eight-screw case, but we may state them, at all events, for the case of n = 3. 



Five screws belonging to a three-system, and their five correspondents 



