Sfi8 On the €o?npositton and Decomposition of Forces, 



coniposants, and is equal to their sum: therefore, the re- 

 sultant of the forces P, Q, is equal to their sum, and its 

 direction, which passes throu";h C, is parallel to BQ or A P. 



2. The right line AB bemg the half of EF, we have 

 AB=EC, and by subtracting from each of these equal 

 quantities thc»part AC, which is common, vve have BCx= 

 EA=AD. 



Also, because AB=CF, by taking away the common 

 part CB, we have AC = BF=BD, and because, bv suppo- 

 sition, P I O : : AD : DB, 

 we shall have P : Q : : BC : AC. 



But the resultant of the two forces P,Q, has been proved 

 to pass through C : hence the point of application of this 

 resultant divides the right line AB into two parts which 

 are reciprocally proportional to the two forces. 



19. Cor, 1. Therefore, to produce an equilibrium with 

 the two forces P, Q, we must divide the right line AB in 

 C, so that the two parts may be reciprocally proportional 

 to these two forces, and apply to the point C a third force 

 equal to the sum P-fO, in a contrary direction but parallel 

 to AP or BQ. 



20. Remark, If the relation of the forces P, O, and 

 the length of the right line AB be given in numbers, and 

 we want to determine the distance of the point C from 

 A or B, the proportion 



P:Q::BC:AC 

 cannot be immediately applied, because in this proportion 

 we only know the two first terms : but since 

 P ; O : : BG : AC, 

 by composition P+g : Q : : BC-hAC=AB : AC; 

 also P + Q: P: : AB :BC; 

 in each of which the three first terms are given. 



21. Cor,^, When a single force R is applied to a poii^t 

 C, in the inflexible right line AB, we can always resolve 

 it into two others P, Q, which being applied to the two 

 given points A, B, and directed parallel to RC, will pro- 

 duce the same effect; for the force R may be divided into 

 two parts which are reciprocally proportional to the line 

 AC, BC, by means of the two following proportions 



AB : BC : : R : P 



AB : AC : : R : O. 

 In each of which we know the three first terms. And 

 the resultant of the two forces P, Q, has the same quantity 

 and direction, imd acts the same way as the force R. 



22. Cor. 3. Every thing being (in fig. 5) as in the pre- 

 ceding corollary, if wc apply to the point C, of the right 



lin« 



