48 Proceedings of the Royal Irish Academy. 



The twist will move the point indicated by a, through the vector /*, + VXm^, 

 and the virtual moment of this displacement and /3, is 



- -S^, (jtr -r TA-a.) = - Vj3. - S\. Va^i. (54) 



In like manner the work done by yS. is 



-S^.ijL.^ FA^O = - Sfi^ - 5A, Fo^,. (55) 



Hence the total work done by the wrench (/*,, Ai) in the course of the twist 



(fuA.) is 



- S^ '/9, + /80 - 5X, ( VaS^ - Va^,) = - 5 (/x.A, - /^A,). (56) 



This expression for the virtual moment of a twist and a wrench may be 



transformed as follows : — 



"We have, in general, 



p = pa'^a (57) 



= (iipa* + Vpa'^ia 



= uSpa' + !>.-' . a. (58) 



We shall now make 



/», = -S'/i.A,-', u, = I'^M.V, (59) 



p, = .ST/ijAr', <.': = V/tX-': (60) 



whenc-e 



/»i = piX, + *>,A„ /J; = ^'jAj -r irfsAi, (61) 



and 



- S{fi,X, + /isA.) = - S{pX + *.,A,) A, - S(}hXt t «u,A,)X, 



=■ - (/»! + Pt) SXiXz + 5(»», - a.,) A, A, 



= - iPi ^ p.) ^X ^ S(,^- w,) I'X.X^ (62) 



We have now to substitute in this expression as follows : — 



SXX. = - n.TA. cosO, (63) 



S{wt - «,)A,X, = - rA,rA,.rf.sinfl, (64) 



where B is the right-handed angle between the two screws, and d the shortest 

 distance between them. 



The first of these equations is obvious. As to the second, we may observe 

 that, without any loss of generality in the expression S(un - (i>,)X,Ai, we may 

 take <i>i and w, to correspond to the points in which the two screws are inter- 

 sected by their common perpendicular, so that u/- - «ki = the product of d into 

 a unit vector directed from screw 1 to screw 2. If the two screws form a 

 right pair, this vector coincides in direction with FA,Ai, and 



5 (*^ - «,) X.X. = s(^ - •,,) r\x 



is a negative quantity, and sin 6 is positive, as of course it should be ; for in 

 this case the right-handed angle is i 180°. But if the two screws form a left 

 pair, then m - wi and FXiAi, though on the same axis, are in opposite directions, 

 and accordingly sin 6 is negative. 



