Ball — Contributions to the Theory of Screws. 



47 



We might have obtaiued this result by immediate solution for p from 

 formula (44), which may be written 



/J. - p\ -^ SpX = p\, (48) 



and multiplying into X-' 



fM\-' - p + \-hS\p = p ; (49) 



taking the vector and denoting the scalar SpX'^ by t we have the desired form. 

 The equation (42) may be written 



Sa = £ + F/t? + Vi (a - v)- (50) 



Thus we see that the displacement of the system may be represented by a 

 small right-handed rotation about the vector i drawn through any point 17 

 if accompanied by the translation e + Virj. 



If /J,, X be each increased in the ratio of a given scalar m so as to become 

 mfi and mX, the pitch SfiX'^ and the vector-perpendicular F/^X"' from the 

 origin on the screw are alike unaltered. Each different value of m cor- 

 responds to one of the singly infinite number of wrenches which may have 

 one and the same screw as their site. 



Of course, where fi, X are both known, not only is the screw determined 

 (which requires 5 data), but also the intensity of the wrench is known. 

 Thus a knowledge of /i and X gives six data, expressing everything about 

 the force system. 



Perhaps the most useful theorem in the application of quaternions to the 

 theory of screws is that which is enunciated as follows* : — 



If a rigid system acted upon b)" a wrench, represented by the pair of 

 vectors (/ai, Ai), receive a small twist, represented by the pair of vectors 

 (/^j, Xa). then the work done is 



- S{fj,iXn + /u.Ai). 



The following proof of this important expression of 

 the virtual moment may be given : — 



A wrench [fi, X) can in an infinite number of ways 

 be replaced by two forces jSi and /3a (fig. 14) acting 

 at points a, and a-, respectively; /3i maybe transferred 

 to the origin with the introduction of the couple repre- 

 sented by the vector Fai/3i. In lilvc manner we can 

 transfer /S2 to the origin with the introduction of the 

 couple Fao/3j. We thus have 



/3, + /3. = X,. (52) 



VaS, + Vd^, = /x,. (53) 



(51) 



V. 



Fio. 14. 



* Joly, in Hamilton's " Elements," vol. ii., p. 390; also "Manual," p. 204. 

 K. I. A. PROO,, VOL. XXVUI., SECT. A. [7] 



