12 MR CLERK MAXWELL ON 



faces are not only rigid, but are capable of internal stress, independent of 

 external forces, and the expression of this stress depends on six independent 

 variables. 



In a polyhedron with triangular faces, if a number of the edges be taken 

 away so as to form a hole with e 1 sides, the number of degrees of freedom is 



p == gj — Qn + 3 . 



Hence, in order to make an n-\j connected polyhedron simply rigid without 

 stress, we may cut out the edges till we have formed a hole having 6 n — 3 edges. 

 The system will then be free from stress, but if any more edges be removed, the 

 system will no longer be rigid. 



Since in the limiting case of the inextensible surface, the smallest hole may 

 be regarded as having an infinite number of sides, the smallest hole made in a 

 closed inextensible surface connected to any degree will destroy its rigidity. 

 Its flexibility, however, may be confined within very narrow limits. 



In the case of a plane frame of s points, we have 2s data required to deter- 

 mine the points with reference to a given origin and axes ; but since 3 arbitrary 

 data are involved in the choice of origin and axis, the number of data required 

 to determine the relative position of s points in a plane is 2s — 3. 



If we know the lengths of e lines joining certain pairs of these points, then 

 in general the number of degrees of freedom of the frame will be 



p = 2s - e - 3 . 



If, however, in any partial system of s 7 points connected by e' hues, the quantity 

 p' = 2s' — e' — 3 be negative, or in other words, if a part of the frame be self- 

 strained, this partial system will contribute only 2s' — 3 equations independent 

 of each other to the complete system, and the whole frame will have p — p' 

 degrees of freedom. 



In a plane frame, consisting of a single sheet, every element of which is 

 triangular, and in which the pieces form three systems of continuous lines, as at 

 p. 11, if the frame contains e pieces connecting s points, s' of which are on the 

 circumference of the frame and s 1 in the interior, then 



3s — s = c + 3 . 



Hence 



p=-(s-s') = -s l , 



a negative quantity, or such a frame is necessarily stiff ; and if any of the points 

 are in the interior of the frame, the frame has as many degrees of constraint as 

 there are interior points — that is, the stresses in each piece will be functions of 

 Sj variables, and s 1 pieces may be removed from the frame without rendering it 

 loose. 



