Ball — On tomographic Screw Systems. 437 



given screw is determined. For from the six equations just written 

 by substitution of known values of ai . . . og and ^i . . . /3s, we can 

 deduce six equations between (11), (12), &c. As, however, the co-or- 

 dinates are homogeneous and their ratios are alone involved, we can 

 only use the ratios of the equations so that each pair of screws gives 

 five relations between the 36 quantities (11), (12), &c. The seven 

 pairs thus give 35 relations which suffice to determine linearly the 

 ratios of the coefficients. The screw /3 corresponding to any other 

 screw a is completely determined ; we have therefore proved that — 



When seven corresponding ^jff//-.s of screws are given, the two hornogra- 

 pJiic screw systems are completely determined. 



A perfectly general way of conceiving two homographic screw 

 systems maybe thus stated: — Decompose a wrench of given intensity on 

 a screw a into wrenches on six arbitrary screws. Multiply the inten- 

 sity of each of the six component wrenches by an arbitrary constant ; 

 construct the wrench on the screw /S 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 (B trace out the homo- 

 graphic screw system. 



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

 twist velocities instead of wrenches. 



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 also 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 ?^-system {Scretos, p. 38), then 

 the w + 1 corresponding screws will also belong to an w-system. 

 If « + 1 screws belong to an w-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, j3, y, 8 are such that suitable 

 wrenches on them, or twist velocities about them, neutraKze. It is 

 then obvious (see Screws, ch. Y.) that each of the determinants must 

 vanish which is formed by taking four columns from the expression — 



o-i. 



tto, 



013, 



O-l, 



do, 



Ofi, 



A, 



A, 



/?3, 



A, 



1^. 



A, 



7i' 



y2» 



73, 



7^' 



7.5, 



76' 



S:, 



82, 



S3, 



S„ 



85, 



8e 



but it is easy to see that these determinants will equally vanish for the 

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



2 2 



