26 Statics. 



imagine the direction of the force p, prolonged till it meets that 

 of the force q in the point A ; AD, AE, being supposed to be 

 the spaces that the forces p, q, can respectively cause the same 

 body to describe, in a determinate time, as one second ; if we 

 form the parallelogram AEID, the diagonal AI will represent the 

 resulting effort of p and q, and may consequently take the place 

 of these two forces. 



Let us conceive now that AI prolonged, meets in B the di- 

 rection of the force r, and having taken BL equal to AI, if we 

 take BF for the space that the force r is capable of making the 

 same body describe in a second ; the force AI being supposed 

 to be applied at B, since this force is represented by BL = AI, 

 from its action combined with the force r = BF, there will re- 

 sult a single force represented by the diagonal BG, of the par- 

 allelogram BLGF. This force, therefore, will take the place 

 of the forces r and AI, that is, it will take the place of the three 

 forces r, q, and p. 



Lastly, let us imagine that BG prolonged, meets in C the direc- 

 ' tion of the force *r, and let us make CK = BG. Let CH represent 

 the space which the force & is capable of making the same 

 given body describe in a second, then by supposing the force 

 BG = CK, applied at C in the direction CG, from the union 

 of this force with the force & there will result a single effort re- 

 presented by the diagonal GAT, of the parallelogram CHNK. 

 This force, therefore, will take the place of the forces zr and CK, 

 or of -SF and BG ; it will consequently take the place of the four 

 forces jo, q, r, w, and is accordingly the resultant of these four forces. 



It is evident, therefore, that any number of forces, when 

 exerted in the same plane, may be reduced to a single force, 

 and the manner in which this may be done is also manifest. 



45. It will be seen moreover from the above example, how 

 we may always substitute for a single force as many others as we 

 please, and what are the requisite conditions for effecting this,. 

 Instead of the single force BG, for example, we may, by form- 

 ing the parallelogram BLGF, of which BG is the diagonal, take 

 the two forces represented by BF, BL ; and as we may suppose 

 each of these two forces applied at any such point of their direc- 

 tions respectively as we choose, we can transfer BL to AI, (the 



