Ball — On ^tomographic Screw Systems. 439 



By the aid of these invariant functions it is easy, when seven pairs 

 of screws are given, to construct the screw con-esponding to any given 

 eighth 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 17 . 68 



These can be all 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 attri- 

 bute of four coUinear points. The curious inquirer may be tempted 

 to speculate on the analogy between a group of eight screws which 

 satisfy one or more of the conditions 



12 . 38 ^ Ti . 28 = 0, 



and a row of four j)oints, whereof two cut the other pair harmo- 

 nically. 



The invariants are also very easily deduced by considerations of a 

 mechanical nature. It is not hard to conceive that to a dyname on 

 one screw corresponds a dyname on the corresponding screw, and that 

 the ratio of the intensities of the two dynames is to be independent 

 of their intensities. AVe 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 consti- 

 tute 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 [5. To one 

 screw a will correspond one screw yS, and (since the body is perfectly 

 free) to one screw /? will correspond one screw a. It follows, from 

 the definition of homography, that as a moves over every screw in 

 space, /? will trace out an homographic system. . . . Prom the laws of 

 motion it will follow, thiit if F be the intensity of the impulsive 

 wrench, and if V be the twist velocity Avhich that wrench evokes, 

 then F ^ Fwill be independent of i^and V, though, of course, it is 

 not independent of the actual position of a and /?. 



It is known [Scretvs, p. 171) 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. 



