176 RECIPROCAL FIGURES, FRAMES, 



multiplied by the distance of the points between which it acts, is equal to 

 the sum of the products of the repulsions multiplied each by the distance of 

 the points between which it acts. 



For since each point is in equilibrium under the action of a system of 

 attractions and repulsions in one plane, it will remain in equilibrium if the 

 system of forces is turned through a right angle in the positive direction. If 

 this operation is performed on the systems of forces acting on all the points, 

 then at the extremities of each line joining two points we have two equal 

 forces at right angles to that line and acting in opposite directions, fornnn<: 

 a couple whose magnitude is the product of the force between the points and 

 their distance, and whose direction is positive if the force be repulsive, and 

 negative if it be attractive. Now since every point is in equilibrium these two 

 systems of couples are in equilibrium, or the sum of the positive couples is 

 equal to that of the negative couples, which proves the theorem. 



In a plane frame, loaded with weights in any manner, and supported by 

 vertical thrusts, each weight must be regarded as attracted towards a horizontal 

 base line, and each support of the frame as repelled from that line. Hence 

 the following rule: 



Multiply each load by the height of the point at which it acts, and each 

 tension by the length of the piece on which it acts, and add all these 

 products together. 



Then multiply the vertical pressures on the supports of the frame each by 

 the height at which it acts, and each pressure by the length of the piece on 

 which it acts, and add the products together. This sum will be equal to the 

 former sum. 



If the thrusts which support the frame are not vertical, their horizontal 

 components must be treated as tensions or pressures borne by the foundations 

 of the structure, or by the earth itself. 



The importance of this theorem to the engineer arises from the circum- 

 stance that the strength of a piece is in general proportional to its section, so 

 that if the strength of each piece is proportional to the stress which it has 

 to bear, its weight will be proportional to the product of the stress multiplied 

 by the length of the piece. Hence these sums of products give an estimate of 

 the total quantity of material which must be used in sustaining tension and 

 pressure respectively. 



