82 Statics. 



two forces q and r ; hence, in order that there may be an equilib- 

 rium, it is necessary that q should be represented by BA, and 

 r by CA, p by supposition being represented by AD. We have, 

 therefore, the following proportions, 



p i q n AD i JIB, and p : r : : AD : AC-, 

 that is, 



p : q : r : : AD : AB : AC. 



Such is the ratio that must exist among the forces p, q, r, ia 

 order that an equilibrium may take place. 



142. Since the two forces q, r, must be equal to the two for- 

 ces AB, AC, which are the components of the force p, we infer 

 that, when there is an equilibrium among three forces, any two 

 of them must have the same ratio to the third, that two compo- 

 nents have to their resultant. 



48. 143. Accordingly we have the proportion, 



p : q : r : : sin BAC : sin CAD : sin DAB 

 :: sin q A r : sin r AS : sin q AS, 



p A being produced toward S ; that is, when three forces are in equi- 

 librium, each is represented by the sine of the angle comprehended be- 

 tween the directions of the two others ; these directions being pro- 

 duced if necessary. 



144. Since the three forces/?, q, r, which are to be in equi- 

 librium, are represented by AD, AB, AC, or, which amounts to 

 the same thing, by the sides AD, AB, BD, of the triangle ABD, 

 of which the angles ABD, EDA, DAB, are equal to the angles 

 CA*q, rAS,qAS, determined by the directions of the forces, 

 it will be seen that all the questions which can occur with re- 

 spect to the value and direction of the forces, requisite to an 

 equilibrium, refer themselves to the subject of trigonometry. If, 

 for example, the values of three forces p, q, r, were given, and 

 it were proposed to find their direction, we should resolve the 

 Trig.38. triangle DBA, the three sides of which would be known, and the 

 angles thus obtained would give the directions of the forces re- 

 quired. If we had given the two forces p, q, and the angle/? A q, 

 of their directions, or its supplement q AS = DAB ; then we 

 should have the two sides AB, AD, and the contained angle DAB, 



