249] HOMOGRAPHIC SCREW SYSTEMS. 265 



seven pairs thus give 35 relations which suffice to determine linearly the 

 ratios of the coefficients. The screw y9 corresponding to any other screw a is 

 completely determined ; we have therefore proved that 



When seven corresponding pairs of screws are given, the two homographic 

 screw systems are completely determined. 



A perfectly general way of conceiving two homographic screw systems 

 may be thus stated : Decompose a wrench of given intensity on a screw a 

 into wrenches on six arbitrary screws. Multiply the intensity of each of the 

 six component wrenches by an arbitrary constant ; construct the wrench on 

 the screw /3 which is the resultant of the six components thus modified; 

 then as a moves into every position in space, and has every fluctuation in 

 pitch, so will /3 trace out the homographic screw system. 



It is easily seen that in this statement we might have spoken of twist 

 velocities instead of wrenches. 



249. Homographic n-systems. 



The seven pairs of screws of which the two systems are defined cannot be 

 always chosen arbitrarily. If, for example, three of the screws were co- 

 cylindroidal, then the three corresponding screws must be co-cylindroidal, 

 and can only be chosen arbitrarily subject to this imperative restriction. 

 More generally we shall now prove that if any n + 1 screws belong to an 

 n -system (69), then the n + 1 corresponding screws will also belong to an 

 n-system. If n + 1 screws belong to an ?i-system it will always be possible to 

 determine the intensities of certain wrenches on the n + 1 screws which when 

 compounded together will equilibrate. The conditions that this shall be 

 possible are easily expressed. Take, for example, n = 3, and suppose that 

 the four screws a, fi, 7, 8 are such that suitable wrenches on them, or twist 

 velocities about them, neutralize. It is then obvious ( 76) that each of the 

 determinants must vanish which is formed by taking four columns from 

 the expression 



i, -,, o 3&amp;gt; a,, a 5 , 2 6 



76 



It is, however, easy to see that these determinants will equally vanish for 

 the corresponding screws in the homographic system ; for if we take as screws of 

 reference the six common screws of the two systems, then we have at once 

 for the co-ordinates of the screw corresponding to a 



(ll)a lf (22)0,, (33) a,, (44) 4 , (55) a., (66)*. 



