174 RECIPROCAL FIGURES, FRAMES, 



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

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



p = e, Gn + 3. 



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

 stress, we may cut out the edges till we have formed a hole having 6>/-:i 

 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 * points, we have 2s data required to 

 determine 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 * points in a plane is 2* 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' points connected by e' lines, 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 inde- 

 pendent of each other to the complete system, and the whole frame will have 

 pp 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. 173, if the frame contains e pieces connecting s points, s' of which arc 

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



3s - s ' = e + 3. 

 Hence p= ( *')= s,, 



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

 pouits 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 he 



