128 STATICS. [214. 



sum of the moments of all the forces about any three axes not 

 in the same plane must also vanish. The moment of a force 

 about an axis must be understood as meaning the moment of its 

 projection on a plane at right angles to the axis with respect to 

 the point of intersection of the axis with the plane. This defi- 

 nition is in accordance with the somewhat vague notion of the 

 moment of a force as representing its " turning effect." For, 

 regarding the force as acting on a rigid body with a fixed axis, 

 the force can be resolved into two components, one parallel, the 

 other perpendicular, to the axis ; the former component does 

 evidently not contribute to the turning effect, which is therefore 

 measured by the moment of the latter alone. 



214. The equations of the central axis (Art. 204) can be found 

 by a transformation of co-ordinates. 



Let the system be reduced for any origin O to its resultant R, 

 whose rectangular components we denote by 



and to the vector //"of its resulting couple with the components 



If a point <9 f whose co-ordinates are f , 77, f be taken as new 

 origin and the co-ordinates of any point with respect to parallel 

 .axes through O' be denoted by x\ y\ z\ we have x^^-x\ 

 z-=%+z'. Substituting these values, we find 



L = 2 [ (17 +/)Z- (?+*') Y]=^Z- & K+2(/Z-*' Y) 



where L' is the ^-component of the couple H* resulting for O' 

 as origin. Similar expressions hold for M and N. The com- 

 ponents of H 1 are therefore 



.and its direction cosines are 



