6o STATICS. [105. 



tersecting Q say in q ; through q a line III parallel to #3. The 

 intersection r of I and III is a point of the resultant R which is 

 therefore obtained in position by drawing through r a line equal 

 and parallel to I 3. 



105. In Fig. 22 the two given parallel forces P, Q were 

 assumed of the same sense. The construction applies, however, 

 equally well to the case when they are of opposite sense. The 

 resultant R will then be found to lie not between P and Q, but 

 outside, on the side of the larger force. The construction fails 

 only when the two given forces are equal and of opposite sense, 

 a case that will be considered later (see Art. 112 and 

 Arts. 128-138). 



106. To determine the position of R analytically, we may find 

 the ratio in which it divides the distance (perpendicular or 

 oblique) between P and Q. Let s (Fig. 22) be the point where 

 R meets pq. Then, since the triangles prs and O i 2, as well as 



the triangles qsr and O 2 3, are similar, we have 



x 



gs_O_2 } sq _O2 . 

 sr~~ P ' sr~~ Q 



hence, dividing, <c = M. 



sq P 



This means that the resultant of two parallel forces divides their 

 distance in the inverse ratio of the forces. As this proposition 

 finds application in the theory of the lever, it is commonly 

 referred to as the principle of the lever. 



Dropping perpendiculars /, q from any point of the resultant 

 R on the components P, Q, the relation can be expressed 

 in the form 



Pp=-Qq, 



which shows that Varignon's proposition of moments (Arts, 

 89-93) applies to parallel forces. 



