AND DIAGRAMS OF FORCES. 171 



when m remains constant, however many lines be drawn. But in the case of a 

 simple closed surface 



Mi- 2, 



If the closed surface is doubly connected, like that of a solid body with a 

 hole through it, then if we draw one closed curve round the hole, and another 

 closed curve through the hole, and round one side of the body, we shall have 

 e = 2, s=l, fl, so that m = 0. If the surface is n-\y connected, like that of 

 a solid with n 1 holes through it, then we may draw n closed curves round 

 the n 1 holes and the outside of the body, and n I other closed curves each 

 through a hole and round the outside of the body. 



We shall then have 4( 1) segments of curves terminating in 2(n 1) 

 points and dividing the surface into two faces, so that e = 4(n 1), s = 2(n 1), 



and /= 2, and 



e s f=2n 4, 



and this is the general relation between the edges, summits, and faces of a 

 polyhedron whose surface is n-ly connected. 



The plane reciprocal diagrams, considered as plane projections of such poly- 

 hedra, have the same relation between the numbers of their lines, points, and 

 polygons. It is manifest that since 



e^ = e a s, =/ and /I = s 2 , 

 where the suffixes refer to the first and second diagrams respectively 



n l = n a , 

 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 3s -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. 



222 



