RECIPROCAL FIGURES, FRAMES, AND DIAGRAMS OF FORCES. 15 



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 1 y 1 z x , 

 x 2 y 2 z 2 , x p y p z p , &c, and let the force between any two points p, q, be ~P pq , 

 and their distance r pq , 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 



(x p - asj)^ + (ocp - x 2 )^ + &c. + (xp - x g )^ + &c. = , 

 t Pi r P2 r pq 



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



P 



x\{(x p -x t )^} = 0. 



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

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



2? 2? { (x P - ^) 2 — 1 = , 

 1 l (. r pt ) 



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



which is the algebraic expression of the theorem. 



General Theory of Diagrams of Stress in Three Dimensions. 

 First Method of Representing Stress in a Body. 



Definition. — A diagram of stress is a figure having such a relation to a 

 body under the action of internal forces, that if a surface A, limited by a closed 

 curve, is drawn in the body, and if the corresponding limited surface a be drawn 

 in the diagram of stress, then the resultant of the actual internal forces on the 

 positive side of the surface A in the body is equal and parallel to the resultant 

 of a uniform normal pressure p acting on the positive side of the surface a in 

 the diagram of stress. 



Let x, y, z be the co-ordinates of any point in the body, £ ??, I those of the 

 corresponding point in the diagram of stress, then £ -q, £ are functions of x, y, z, 

 the nature of which we have to ascertain, so that the internal forces in the body 

 may be in equilibrium. For the present we suppose no external forces, such 

 as gravity, to act on the particles of the body. We shall consider such forces 

 afterwards. 



Theorem 1. — If any closed surface is described in the body, and if the stress 

 on any element of that surface is equal and parallel to the pressure on the cor- 



