10 MR CLERK MAXWELL ON 



polyhedra, have the same relation between the numbers of their lines, points, 

 and polygons. It is manifest that since 



h = e 2 , s 1 =f 1 , and f x = s 2 , 



where the suffixes refer to the first and second diagrams respectively 



or the two diagrams are connected to the same degree. 



On the Degrees of Freedom and Constraint of Frames. 



To determine the positions of s points in space, with reference to a given 

 origin and given axes, 3s data are required; but since the position of the origin 

 and axes involve 6 data, the number of data required to determine the relative 

 position of s points is 3s — 6. 



If, therefore, the lengths of Ss — 6 lines joining selected pairs of a system of 

 s points be given, and if these lengths are all independent of each other, then 

 the distances between any other pair of points will be determinate, and the 

 system will be rigidly connected. 



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

 a system of s' of the points are more than 3s' — 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 

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

 by e' 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 

 degrees 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 

 degrees of freedom of the complete system are p — p', where p and p' are got 

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

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

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



We have also 



mf = 2e . 

 e — s - f = 2n — 4, 



and 





whence 



3s — e = p + 6 , 





P = 6(1 - n) + (2 - £) 



