Ball — On tomographic Sarew Systems. 445 



three screws reciprocal to A, then the single screw jB, which is reci- 

 procal to the five screws thus found, belongs to both A and B. We 

 thus see that to each screw a of ^, one corresponding screw in the same 

 system can be determined. The result just arrived at can be similarly 

 shown generally, and thus we find that when every screw in space 

 corresponds to a screw of an w-system, then each screw of the n-system 

 will correspond to a ( 7 - ?i) system, and among the screws of this 

 system one can always be found which lies on the original ?i-system. 



As a mechanical illustration of this result we may refer to the 

 theorem, that if a rigid body has freedom of the ?^"' order, then, no 

 matter what be the system of forces which act upon it, we may always 

 combine the resultant wrench with certain reactions of the constraints, 

 so as to produce a wrench on a screw of the n-system which defines the 

 freedom of the body, and this wrench will be dynamically equivalent 

 to the given system of forces. 



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

 shall take a 3-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 

 3-system, then the co-ordinates of any screw on that system will be 

 ttj, a2, 03, the other three co-ordinates being equal to zero. The co- 

 ordinates of the corresponding screw ^ must be indeterminate, for 

 any screw of a 4-system will correspond to /?. This provision is 

 secured by ^84, /Jg, /?6 remaining quite arbitrary, while we have for 

 /?!, IS^, /?3 the definite values, 



^i = (ll)ai + (12)a. + (13)a3, 

 )82 = (21) a, + (22) ao + (23) a„ 

 yS3 = (31)ai + (32)ao + (33)a3. 



If we take ^4, [3^, /?6, all zero, then the values of ^1, /?2, /?3, just 

 written, give the co-ordinates of the special screw belonging to the 

 3-system, which is among those which correspond to a. 



As a moves on the 3-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 



^1 = po-i ; ^2 == po.2 ; (Sz = pas, 



"whence we obtain a cubic for p. 



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

 so that each screw shall coincide with its correspondent. As a dyna- 

 mical 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 of which the body will commence to twist when it 



