AND DIAGRAMS OF FORCES. 177 



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 to suppose the points all in one plane. This demonstration 

 is mathematically equivalent to the following algebraical proof: 



Let the co-ordinates of the n different points of the system be x^z u 

 X3jf a x p y p z p , &c., and let the force between any two points p, q, be P pq , 

 and their distance r m , and let it be reckoned positive when it is a pressure, 

 and negative when it is a tension, then the equation of equilibrium of any 

 point p with respect to forces parallel to x is 



(Xp-Xj -& + (x p -X a ) -^-f&C. + (x p -X q )-^ + &C. = 0, 

 r pi r pt r pq 



or generally, giving t all values from 1 to n, 



Multiply this equation by x f . There are n such equations, so that if each is 

 multiplied by its proper co-ordinate and the sum taken, we get 



P 



and adding the corresponding equations in y and z, we get 



1 1 

 which is the algebraic expression of the theorem. 



VOL. ii. 23 



