48 STATICS. [87. 



at the same point and lying in the same plane with it. These 

 components are together equivalent to the force, i.e. they may 

 be substituted for the force. 



87. It follows from Art. 85 that the resultant R of two 

 intersecting forces P and Q, including the angle 6, is 



For two parallel forces or two forces acting in the same line, 

 0=o or 1 80, according as they are of equal or opposite sense; 

 hence R = P+Q in the former case, and R = 1?Q in the latter. 

 It is also apparent that the resultant of any number of parallel 

 forces or of forces acting in the same line is found as the 

 algebraic sum of these forces. How the position of the resultant 

 is found in the case of parallel forces will be shown later (Arts. 

 104, 1 06). 



88. By Art. 86, to resolve a force R (Fig. 16) into two com- 

 ponents P y Q along two lines making the angles , /3 with the 

 line of R y we have only to draw through the ends of a vector 



2$=R lines 2 i, 3 i making angles a, fi with 2 3 ; then 2 I =P, 

 i 3 = Q. The triangle 123 gives the relations 



P = Q = R 

 sin/3 sin a sin(a + /3) 



When the components are at right angles, we have P = R cos a, 

 Q = R since. 



89. The projection of a closed polygon on any line being 

 evidently zero, and the resultant being by definition the geo- 



