1 14-] PARALLEL FORCES. 5 



given forces are equal and of opposite sense. In this case, the 

 lines I and III of the funicular polygon become parallel, so 

 that their intersection r lies at infinity. The magnitude of the 

 resultant is of course =o. 



The combination of two equal and opposite parallel forces 

 (F, F) is called a couple. A couple is, therefore, properly speak- 

 ing, not equivalent to a single force, although it may be said to 

 be equivalent to a force of magnitude o at an infinite distance. 

 The theory of couples will be considered in detail in Arts. 128 

 138- 



113. Conditions of Equilibrium. We have seen (Art. 109) that 

 a system of n parallel forces is, in general, equivalent to a single 

 force ; but, as appears from the preceding article, it may happen 

 to reduce to a couple. It follows that for the equilibrium of a 

 system of parallel forces the condition R = o, though always neces- 

 sary, is not sufficient. 



Now, if the resultant R of the n parallel forces F v F%, ... F n be 

 =o, the resultant R 1 of the n I forces F v F 2 , ... F n _^ cannot be 

 o, and its point of application is found (by Art. 1 10) from 

 x = (Ffa + F^ 2 -\ ----- h ^-i-^n-i) /(F 1 + F z -\ ----- h /V-i) and similar ex- 

 pressions for y and z. The whole system of parallel forces is 

 therefore equivalent to the two parallel forces R' and F n . Two 

 such forces can be in equilibrium only when they lie in the 

 same straight line ; i.e. F n must coincide with R' and must 

 therefore pass through the point (x, y, z), which is a point of R '. 



The additional condition of equilibrium is, therefore, 



cosa osc/3 



where a, /3, 7 are the angles made by the direction of the forces 

 with the axes. 



114. For practical application it is usually best to replace the 

 last condition by taking moments about a convenient point. 



PART II 5 



