172 RECIPROCAL FIGURES, FRAMES, 



If, however, the lines are so chosen that those which join pairs of points of 

 a system of ' of the points are more than 3*' -6 in number, the lengths of 

 these lines will not be independent of each other, and the lines of this partial 

 system will only give 3s' 6 independent data to determine the complete system. 



In a system of s points joined by e lines, there will in general be 3s 6 e 

 degrees of freedom, provided that in every partial system of s' points joined 

 by t lines, and having in itself p' degrees of freedom, p' is not negative. If in 

 any such system p is negative, we may put q = p, and call q the number of 

 lingr Bi ii i of constraint, and there will be q equations connecting the lengths of 

 the lines ; and if the system is a material one, the stress along each piece will 

 be a function of q independent variables. Such a system may be said to have 

 q degrees of constraint. If p' is negative in any partial system, then the de- 

 grees of freedom of the complete system are pp', where p and p' are got 

 from the number of points and lines in the complete and partial systems. If . 

 points are connected by e lines, so as to form a polyhedron of f faces, enclosing 

 a space n times connected, and if each of the faces has m sides, then 



mf=2e. 



We have also e s f=2n 4, 



and 3s-e=p + Q, 



(c \ 

 2 -- }e. 

 m/ 



If all the faces of the polyhedron are triangles, m = 3, and we have 



If n=l, or in the case of a simply connected polyhedron with triangular 

 faces, p = o, that is to say, such a figure is a rigid system, which would be no 

 longer rigid if any one of its lines were wanting. In such a figure, if made of 

 material rods forming a closed web of triangles, the tensions and pressures in 

 the rods would be completely determined by the external forces applied to the 

 figure, and if there were no external force, there would be no stress in the rods. 



In a closed surface of any kind, if we cover the surface* with a system 

 of curves which do not intersect each other, and if we draw another system 



* On the Bending of Surfaces, by J. Clerk Maxwell. Cambridge Transactions, 1856. [Vol. i. p. 80.] 



