57, 58, 59, 59 a] VECTOR -COUPLES. 205 



perpendicular distance between their lines, or the arm of the couple. 

 This product is called the moment of the couple. 



Two couples whose planes are parallel give rise to parallel 

 translations, and if their moments are equal, to equal translations. 

 Therefore a rotation -couple may be displaced without altering its 

 effect, if its plane is kept parallel to itself and its moment is un- 

 changed. 



A vector -couple may then be represented by a single vector 

 perpendicular to its plane, whose length is equal to the moment of 

 the couple. Its direction will be governed by the same convention 

 as before, namely, the vector moment is to be drawn in such a 

 direction that rotation in the direction of the couple and translation 

 in that of the moment correspond to the motion of a right-handed 

 screw. 



Moments will be represented by heavy vectors. The moment of 

 a vector -couple is a free vector, hence the composition of couples is 

 simpler than that of the slide -vectors themselves. 



We may now state the theorem of the general infinitely small 

 displacement of a body as follows: The infinitely small displacement 

 of a body may be reduced to a translation and a rotation, or in other 

 words to a rotation and a rotation- couple. The choice of components 

 may be made in an infinite number of ways. 



59. Statics of a Rigid Body. Two equal, parallel, opposi- 

 tely directed forces applied to a rigid body in the same line are in 

 equilibrium. For otherwise they can produce only distortion or 

 motion. Distortion is excluded according to the definition of a rigid 

 body. They satisfy the conditions of equilibrium, 32, for if applied 

 at the center of mass they are in equilibrium, and their moments 

 about any point are equal and opposite. Accordingly a force applied 

 to a rigid body may be applied at any point in its line of direction 

 without change of effect. Thus forces applied to a rigid body are 

 not free, but are sliding vectors (five coordinates). (This is not a 

 property of forces, but of rigid bodies.) Forces, whose lines of 

 direction intersect, may be applied at the point of intersection and 

 compounded by the rule of vector addition. 



AP $ 

 59 a. Parallel Forces. Force - couples. Let A&- and ?*$- 



(Fig. 49) represent two parallel forces applied to a rigid body at A 

 and B. Introduce at A and B two equal and opposite forces AE 

 and BS of any magnitude in the line AB. These being in equili- 

 brium do not affect the system. Find the resultant of AP and AE 

 by the parallelogram, giving AC, also of BQ and BS giving BD. 

 All these forces are coplanar, therefore the lines AC and BD will 



