analogous to the Virial Theorem. 211 



of the forces R (considered positive when repellent) which 

 act mutually between two particles along their line of 

 junction p. If x, y, z and .*'', //, z' be the coordinates of the 

 particles, we have so far as regards the above-mentioned 

 forces, 



X(x'-x) + X{y'-y) + Z(.:'- :) = Rp ; 



or with summation- over ---every pair of -particles SRp. The 

 complete equation may now be written 



2(Xx + Yy + Zz)+XRp = Q, f . . ■ . (2) 



where in the first summation X, Y, Z represent the com- 

 ponents of the fxternaljoreez operative at the point x, y, z. 

 Jn Maxwell's example the only external forces are the weights 

 of the various parts of the system (supposed to be concen- 

 trated at the junctions of the struts and ties), and the reactions 

 at the foundations. 



The analogous theorem, to which attention is now called, 

 is derived in a similar manner from the equally evident 

 equation 



Z[x.%Y + y.2X]=() (3) 



We have to extract from the summation on the left the force 

 R mutually operative between the particles at a, y, z and at 

 ■/•', y f , z' '; and we shall limit ourselves to the case of two 

 dimensions. If X, Y be the components of force acting 

 upon the latter particle, p the distance between the particles, 

 and the inclination of p to the axis of x, we have 



Y(x'- x) + X< y'-y) = Rp sin 20 ; 



so that if now X, Y represent the total external force acting 

 at x, y, (3) becomes 



2|>Y+^X]+£Bpsin20=6. . . . (4) 



where the first summation extends to every particle and the 

 second to every pair of particles. 



If the external force at x, y be P and be inclined at an 

 angle a, we have 



X = Pcosa, Y = Psina; 



so that, if x = r cos 0, y — r sin 6 as usual, (4) may be written 



t Pr sin -(0+«)+2R/> sip 20=0. . . (5) 



As simple examples of these equations, consider the square 

 framework with one diagonal represented in figs. 1 and 2, 

 and take the coordinate axes parallel to the sides of the square. 

 Since sin 20 = for all four sides of the square, the only R 



