28 Proceedings of the Royal Irish Academy. 



From this we can prove the following fundamental theorem : — 

 If twists of amplitudes a, /3', 7' about three vector-screws a, j3, 7 neutralize 

 each other when applied to the same rigid body, then wrenches of intensities 

 a", /3", y" on the same vector-screws «, /3, j will be in equilibrium when 

 applied to a rigid body of 



,." : /3" : 7" : : ■•' : ^' = l'- (5) 



This is a consequence of the symmetry of the virtual coellicient of two 

 vector-screws with regard to these vector-screws;* for, from the condition 

 stated (5), the equation (4) may be written 



a'-^ar, + i3"wp, + 7"ct„ = 0. (6) 



If »)' be the amplitude of a small twist about n, then (6) may be expressed 

 thus 



2,'a"ta., 4. 2.,73"np, + 2„'7"^v. = 0- (7) 



This shows that three wrenches of intensities a", /j", 7" on the vector-screws 

 n, /3, 7 do collectively no work when the body receives a twist about any 

 screw whatever. It follows that three wrenches must equilibrate, and the 

 desired theorem has been proved. 



It thus appears that twists and wrenches are comjJounded by laws which 

 can be derived from (4) and (6) by merely attributing to 1/ various jiositions 

 and pitches.t Wo may commence by showing that wlien wrenches of inten- 

 sities 11", /3", 7" respectively on three vector -screws a, /3, 7 equilibrate, tlicn 

 the line intereecting two of those screws perpendicularly must also intersect 

 the third pei-pendicularly. 



We observe that tlie virtual coellicient of two screws wliicli intersect at 

 right angles is zero; for then botli f/ = and cos = 0. If therefore wo take 

 for I) any screw on the common perpendicular intersecting a and /3, we have 



ro., = and w^, = 0, 

 and therefore from (6) 



We cannot satisfy this by making 7" = ; for then the two wrenches on 

 o and /3 would have to equilibrate, which is not possible unless o and /3 are 

 identical screws : rejecting this case, we have 



If as usual d is the shortest distance between 7 ami ij, and fj Lhu right- 

 handed angle between them, we infer that 



(Py + Vr,) C03 9 - d sin = 0. 



• Klein, Math. Ann., vol. iv., p. 413 (1871). 



t Of course these laws are already well known (" Treatise," p. 18), but their derivation from 

 formula) (4) nnd (6) will be useful in what foUowF 



