Oa tlie Composition and Decomposition of Forces, 337 



P, O, R, S, &c. whose directions comprised in the same 

 plane do not concur in the same point; whose points of 

 apphcation A, B, C, D, &c. are connected together in an 

 invariable manner; and whose magnitudes are represented 

 by the parts Aa, Bb^ Cc, Dd, of their directions. 



Solution. After having prolonged the directions of 

 any two of these forces, such as P, O, until ihey have met 

 somewhere in a point E; we lay off from E to F and from 

 E to G the hnes Aa, B/;, representinjr these forties, which 

 will complete the parallelogram EFeG, whose diagonal Ke 

 will represent in magnitude and direction the resultant T 

 of the two forces Pj'Q, (36). 



Instead of the forces Pj Q, we take the resultant T, and 

 prolong its direction as well as that of the force R until 

 ihey meet somewhere in a point H; we lay off the line 

 Ee from H to I, and the line Cc" from H to K ; which will 

 complete the parallelogram HI/iK, whose diagonal Hk 

 will represent in magnitude and direction the resultant V 

 of the two fwrces T, R, which will also be that of the 

 three forces P,Q, R. 



In the same manner, instead of the three forces P, Q, R, 

 we take their resultant V, and prolong its direction as. 

 well as that of the force S until they meet in a point L; 

 then laying off from L to M and from L to N the lines 

 HA, Dd, which represent the forces Vand S, they complete 

 the parallelogram LM/N, whose* diagonal hi will represent 

 the resultant X of these two forces, which is also that of 

 the four forces P, Q, H, S, 



By proceeding thus we may find the magnitude and di- 

 rection of the general resultant of all the proposed forces, 

 whatever may be their number. 



42. Coj\ Therefore when several forces, directed in the 

 same plane, are applied to points connected together in an 

 invariable manner, these forces have always a resultant; so 

 that it is possible to make them in equilibrium by means 

 of one force only ; except in the case where the direction of* 

 one of these forces being parallel lo that of the resultant 

 of all the others, this force and this resultant are equal to 

 each other, and act in contrary directions : for we have seen 

 (24) that then to make them in equilibrium it is necessary 

 to apply a force of nothing, whose direction should pass 

 to an infinite distance; which is impracticable. 



THEOREM. 



43. If three forces P, Q, R, have their magnitudes and 

 directions represented by the three edges AB, At, AD, con- 



Vol. 35. No. 145.. May 1810, Y tigiious 



