104 STATICS OF EIGID -BODIES 



COUPLES IN PARALLEL PLANES 



80. The result of 79 shows that two couples acting in the 

 same plane produce the same effect if their moments are equal. 

 For, on reversing one of them, all the conditions of equilibrium 

 are satisfied. 



Thus we can determine the effect of a couple in any plane by 

 knowing its moment only. We shall now show that the actual 

 plane in which the couple acts is immaterial, the direction of this 

 plane alone being of importance. In other words : 



Couples of equal moments acting in parallel planes produce the 

 same effect. 



To prove this, we reverse one couple and show that the two 

 couples are then in equilibrium. Let the first couple consist of 

 I? v two forces each equal to R, 



and let a common perpen- 

 dicular to their lines of 

 action meet the latter in 

 points A, B. Let A'B' be 

 a line equal and parallel 

 to AB in the plane of the 

 second couple, and let the 

 R second couple reversed be 



represented by two forces 



R, R at A'B'. We can regard this couple as representing the second 

 couple reversed, for its moment is equal and opposite to that of 

 the second couple, while it is acting in the same plane as the 

 second couple. 



We have now to show that the four forces each equal to R 

 acting at A, B, A', B' are in equilibrium. By construction, ABB' A' is 

 a parallelogram, so that C, the point of intersection of its diagonals, 

 is also the middle point of each diagonal. 



The two parallel forces R, R acting at A, B 1 may be compounded 

 into a single force 2 R acting at (7, the middle point of AB', and 

 similarly the two forces R, R at B, A' may be compounded into 



