40 FORCES ACTING ON A SINGLE P ARTICLE 



As this is a particular case of the polygon of forces no separate 

 proof is needed. As before, the converse is also true, so that the 

 condition is a necessary and sufficient condition for equilibrium. 



When there are only three forces acting, the condition for equi- 

 librium can be expressed in a still simpler form : 



32. LAMI'S THEOREM. When a particle is acted on by three 

 forces, the necessary and sufficient condition for equilibrium is that 

 the three forces shall be in one plane and that each force shall be 

 proportional to the sine of the angle between the other two. 



Suppose that a particle is acted on by three forces P, Q, R. The 

 necessary and sufficient condition for equilibrium is that we can 

 form a triangle by placing end to end three 

 lines which represent the forces P, Q, R in 



magnitude and direction. 



Let us begin by taking AB to represent 

 P, and placing against it at B a line BC to 

 represent Q. Then CA must represent R, if 

 the conditions for equilibrium are to be satis- 

 fied. Thus the three forces must be in one 

 plane, namely, the plane parallel to ABC 

 through the point of action of the forces. 



Assume that there is equilibrium, so that 

 the three forces are represented by the sides of 

 the triangle ABC. Let us denote the angles of the triangle as usual 

 by A, B, C, and its sides by a, b, c. Then, from a known property 



of the triangle, 7 



a o c . 



sin A sin B sin C 



By our construction, however, a, b, c are proportional to the 

 magnitudes of the forces : we have 



c b a 



Thus P * 



sin C sin A sin B 



