14 MR CLERK MAXWELL ON 



angles to that line and acting in opposite directions, forming 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 acid 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. 



The following method of demonstrating this theorem does not require the 

 consideration of couples, and is applicable to frames in three dimensions. 



Let the system of points be caused to contract, always remaining similar 

 to its original form, and with its pieces similarly situated, and let the same forces 

 continue to act upon it during this operation, so that every point is always in 

 equilibrium under the same system of forces, and therefore no work is done by 

 the system of forces as a whole. 



Let the contraction proceed till the system is reduced to a point. Then the 

 work done by each tension is equal to the product of that tension by the distance 

 through which it has acted, namely, the original distance between the points. 

 Also the work spent in overcoming each pressure is the product of that pressure 

 by the original distance of the points between which it acts ; and since no work 

 is gained or lost on the whole, the sum of the first set of products must be 

 equal to the sum of the second set. In this demonstration it is not necessary 



