On the Composition <tnd Decomposition of Forces. 335 



mains only to show that its quantity will be represented by 

 this diagonal. To the point A apply a force S equal and 

 directly" opposite to the resultant U, this force will be di- 

 rected according to the prolongation of the diagonal DA, 

 and the three forces P, Q, S, will be in equilibrio. There- 

 fore the force O, will also be equal and directly opposite 

 to the resultant of the two other forces P, S ; and conse- 

 <juently this last resultant will be directed according to the 

 .prolongation of the right line CA. Produce CA till All 

 ==AC, and draw thn right line HB, which will be parallel 

 to AD, and consequently to the direction of the force S, 

 4lso from H draw HK parallel to the clij-ection of the force 

 P j the two forces P, S, are proportional to the sides AK, 



AB, of the parallelogram ABE^K, (34) ; that is, we shall 

 have P : S : : AB : AK or HB. 



Now because ADBH is a parallelogram, we have HB = 

 AD J moreover the forces S and R are equal, therefore 



P : R : : AB ; AD. 

 But by supposition P : O : : AB : AC ; 

 tl^erefore, by uniting these two proportions, we get 

 P : Q ; R : : AB : AC : AD. 



37. Cor. 1. If the two forces P, Q, are applied to the 

 point A, they will be in equilibrio with a third force applied 

 to the same point, the direction of which is DA, and 

 which is proportional to the diagonal AD ; for this force 

 will be equal and directly opposite to the resultant of the 

 forces P, Q. 



If the forces P, O, are npplied to any other points of 

 their directions, there will be an equilibrium in applying to 

 any point whatever of the right line AD, and in the di- 

 rection DA, a force proportional to AD ; provided that 

 the point of application of this last force be invariably con- 

 nected to the points of application of the forces P, O. 



3S. Cor. 2. VVe can always decompose a force R, given 

 in quantity and direction, into two other forces P, O, di- 

 rected accorduig to the given right lines AP, AQ ; pro- 

 vided that these directions and that of the force R, are 

 comprised in the same plane, and concn.ir in the samp 

 point A. 



For let the force R be represented bv a part AD of its 

 direction, and drawing, through the point D, the rioht 

 lines DC, DB, parallel lo the given dirtctions AP, AQ, they 

 form the parallelogram ABDC, the sides of whicli AB, 



AC, represent the forces P, O, required ; for (30) the re- 

 sultant of these forccB will have the same quantity and di- 

 rection 



