338 On the Composition and Decomposition of Forces. 



tiguous to the same angle of a paralleiopipedon ABFEGD, 

 in such a way that P : Q : R : ; AB : AG t AD, their re- 

 sultant S will be represented in magnitude and direction by 

 the diagonal AE of the parallelopipedon contiguous to that 

 angle, and we shall have P : O : R ; S : : AB^: AC : AD : 

 AE. 



Demonstration. On the face ABFC, which contains 

 the directions of the two forces P, Q, let the diagonal 

 AF be drawn ; let the diagonal DE also be drawn on the 

 opposite face DHEG; these two diagonals are parallel and 

 equal, for the two edges AD, EF, of the parallelopipedon, 

 at the extremities of which they terminate, are parallel and 

 equal; therefore AFED will be a parallelogram. Hence 

 the two forces P, Q, being represented in magnitude and 

 direction by the sides AB, AC, of the face ABFG, which 

 is 3 parallelogram, their resultant T will be represented in 

 magnitude and direction by the diagonal AF, and we shall 

 liave P: g :T: : AB : AC : AF. [ 



In like mainier the two forces T, R, being represented 

 by the sides A F, AD, of the parallelogram AFED, their 

 resultant S, which is also that of the three forces P, Q, R, 

 will be represented by the diagonal AE of the same paral- 

 lelogram, and we have T : R : S : : AF : AD : AE; there- 

 fore, by uniting the above two series of proportionals we 

 have P : Q : R : S : : AB : AC : AD : AE. 



Now the diagonal AF' is likewise that of the paral- 

 lelopipedon ; therefore the resultant of three forces will be 

 represented in magnitude and direction by the diagonal of 

 the parallelopipedon. 



44, Cor. \Ve may always decompose a force S civen 

 in magnitude and direction into three other forces P, Q,R, 

 directed according to the three given lines AP, AO, AR, 

 not comprised in the same plane, provided these three di- 

 rections and that of the force S concur in the same point A. 

 To effect this, by the three directions considered two 

 and two we draw the three planes BAC, CAD, DAB ; 

 he force S will be represented by a part AE of its direction ; 

 and by the point E we draw three other planes, EGDH, 

 EHBF, EFCG, respectively parallel to the three first; these 

 six planes are the faces of a parallelopipedon, whose dia- 

 gonal is AE, and whose edges AB, AC, AD, which arc 

 taken on the three given directions, represent the niagni- 

 tude of the forces required, P, Q, R ; for (43) the resultant 

 of these three forces will have the same magnitude and 

 direction as the force S. 



Otherwise, we draw through the point E three right 



lines 



