59 a, 60, 61] KINEMATICAL AND DYNAMICAL DUALISM. 



209 



6O. Reduction of Groups of Forces. Dualism. Suppose we 

 have any number of forces applied to various points of a rigid body. 

 Let one such be P applied at A. At any point 

 apply two equal and opposite forces equal and 

 parallel to P. One of these P 2 forms a couple 

 with P. The other is equal and parallel to P. 

 The moment of the couple is perpendicular to 

 this force. 



In this manner the points of application 

 of all the forces may be brought to 0, where 

 they can then be compounded into a single 

 resultant E. For each force thus transferred 

 there remains a couple, and all the couples 

 may be compounded into a single one. There- 

 fore all the forces applied to a rigid body may be replaced by a 

 single force and a single couple. 



We may now state the following dualism existing between 

 infinitesimal rotations and forces: 



Fig. 55. 



Infinitesimal rotations are slid- 

 ing vectors. 



Forces applied to a rigid body 

 are sliding vectors. 



When their axes intersect they are compounded by the vector law. 

 Parallel infinitesimal rotations I Parallel forces 



have a resultant parallel and equal 

 the center of mass of their points 



Two equal and opposite parallel 

 rotations form a rotation - 

 couple represented by its 

 moment, a free vector. 



Every displacement of a rigid 

 body may be reduced to a 

 rotation and a rotation - 

 couple. 



The theory of couples is due 



to their algebraic sum, placed at 

 of application. 



Two equal and opposite parallel 

 forces form a couple, re- 

 presented by its moment, a 

 free vector. 



Every combination of forces 

 applied to a rigid body may 

 be reduced to a force and 

 force -couple. 



to Poinsot. 



61. Variation of the Elements of the Reduction. Central 

 Axis. Null - System. We have seen that any system of slide- 

 vectors may be reduced to the resultant of a single vector and a 

 single moment applied at any point whatever. We have now to 

 examine the variation of the pair of elements, vector E and moment S, 

 as we vary the point of application 0. E is invariable. As we move 

 along the line of E there is no change since E may be applied at 

 any point of its axis, and S may be moved parallel to itself. If we 



WEBSTER, Dynamics. 14 



