Ball — Contributions to the Theory of Screws. 



45 



screw, is at ouco deduced* from Haiiiillmi's I'niKlauuiiital tlieoveiii, expressed 

 in (42), which cau be written in the form 



(( + 8« = a + iSii'^ + Vi {a - Vii-^). 



(43) 



This shows that the displacement of a point P (fig. 13) is produced 

 by a rotation of the system through 

 a right-handed angle Ti, round tlie 

 line of which the equation is 



p = Ffi-i + (i, 

 (where t is a variable scalar) accom- 

 panied by a translation parallel to 

 i and equal to Sti'K 



The following fundamental prin- 

 ciple is well known in quaternionsf: — 



Let /3 and - /3 be a pair of vectors 

 coincident with the lines of action 

 of the two forces of a couple PF' and 

 QQ' respectively. Let T(5 be the 

 magnitude of the force on PP' or 

 on Q(/. Let a be a vector drawn 

 from any point on PP" to any point 

 on QQ', then the couple is completely 



represented by Fj3a. For the couple is right-handed about Fj3a, and TVjia is 

 the moment of the couple. Thus everything about the couple is expressed 

 by Fj3a. 



Wliatever be the forces applied to a rigid body, they may be completely 

 expressed with regard to a given origin by two vectors X, fi. The first 

 vector A represents the resultant of all the forces when transferred in 

 parallel directions to 0. The second vector fj. expresses the resultant of all 

 the couples introduced by transferring the forces to 0. 



If (fi, X) be the vector-moment and force of a system of forces with 

 respect to a point 0, then at a point 0', such that 00' = p, the vector- 

 moment and force of the same system will be p. + VXp, A. 



The expressions just obtained lead in the simplest and most direct 

 manner to the conception of the central axis, and from thence to the 

 foundations of the Theory of Screws. 



Fia. 13. 



* See Joly, in Hamilton's "Elements," vol. ii., p. 390. 

 t Hamilton'3 " Elements," vol. ii., p. 281. 



