252] HOMOGRAPHIC SCREW SYSTEMS. 2G7 



screw. We may solve this problem in various ways. One of the simplest 

 will be to write the five invariants 



12.38 13.48 14\58 15.68 16.78 

 13.28 14.38 15.48 16.58 IT . 68* 



These can be computed from the given eight screws of one system ; hence 

 we have five linear equations to determine the ratios of the coefficients of the 

 required eighth screw of the other system. 



It would seem that of all the invariants of eight screws, five alone can 

 be independent. These five invariants are attributes of the eight-screw 

 system, in the same way that the anharmonic ratio is an attribute of four 

 collinear points. 



251. A Physical Correspondence. 



The invariants are also easily illustrated by considerations of a me 

 chanical nature. To a wrench on one screw corresponds a twist on the 

 corresponding screw, and the ratio of the intensities of the wrench and twist 

 is to be independent of those intensities. We may take a particular case to 

 illustrate the argument : Suppose a free rigid body to be at rest. If that 

 body be acted upon by an impulsive system of forces, those forces will 

 constitute a wrench on a certain screw a. In consequence of these forces the 

 body will commence to move, and its instantaneous motion cannot be 

 different from a twist velocity about some other screw /3. To one screw a 

 will correspond one screw 0, and (since the body is perfectly free) to one 

 screw /3 will correspond one screw a. It follows, from the definition of homo- 

 graphy, that as a. moves over every screw in space, ft will trace out an homo- 

 graphic system.... From the laws of motion it will follow, that if F be the 

 intensity of the impulsive wrench, and if V be the twist velocity which that 

 wrench evokes, then F~ V will be independent of F and V, though, of course, 

 it is not independent of the actual position of a and /3. 



252. Impulsive and Instantaneous Systems. 



It is known ( 230) that when seven wrenches equilibrate (or when 

 seven twist velocities neutralize), the intensity of the wrench (or the twist 

 velocity) on any one screw must be proportional to the sexiant of the six non- 

 corresponding screws. 



Let F 1S , F., s , ... F 7S be the intensities of seven impulsive wrenches on the 

 screws 1, 2, ... 7, which equilibrate, then we must have 



18 28 78 



