Or PABALLEL BOTATIONS. 93 



single Resultant Force, acting at some assigned point as origin, and 

 to a single resultant couple ; the former of these remaining inva- 

 riable, both in magnitude and direction, whatever origin be assumed 

 while the. latter varies in both respects for different origins, remain- 

 ing constant, however, for origins situated along the direction of the 

 Resultant Force. 



Adopting the usual notation by taking as the type of the Forces 

 the rectangular components X, T, Z of the force acting at the 

 point (x, y, z), we have as resultants at the origin of co-ordinates 

 the single Force whose rectangular components are % (X), 2 (T) 

 2 (Z) ; and the single couple whose momental components round 

 the same axes are 



L=2 (Zy— Yz), M=2 (Xz— Zx), N=2 (Yx— Xy). 



If we now seek the resultants corresponding to an origin whose 

 co-ordinates are (V, y' , z'), we find the same Resultant Force, and a 

 new resultant couple (L', M', N'), where 



L' = L + 2(Y).z' — 2(Z)y 

 M' = M + %(Z).x' — 2(X).z' 

 N' = N + ?(X)y — 2(Y).x' 

 From these equations we have 

 L' . 2(X) + M' . S(Y) + N' . S(Z) = L. S(X) + M. %(Y) + N. 2(Z) 

 Hence if the resultant couple be resolved into two whose axes are 

 respectively perpendicular and parallel to the direction of the Re- 

 sultant Force, the latter remains invariable in magnitude whatever 

 origin be adopted ; and hence also the resultant couple will be the 

 least possible when the origin is so assumed that the former vanishes, 

 or, in other words, when the axis of the couple is in the direction 

 of the Force. 



If we seek an origin which shall make the resultant couple 

 vanish, or which shall cause the system of Forces to be reduced to a 

 single resultant Force, we must have for the determination of this 

 origin (x',y',z'), 



L' = o, M' = o, N' = o, 

 or 



o = L + 5(Y>' - S(Z)y ) 



o = m + s(zy - 2(xy } (i) 



o = N + 2(Xy - S(Y)#' ) 

 These equations are inconsistent unless a certain condition hold, 

 which is, 



o = L. S(X) + M. S(Y) + N. 2(Z) (2) 



