TRANSFORMATION OF SURFACES BY BENDING. 87 



the conditions thus obtained would be equivalent to three equations between 

 the six angles of the edges belonging to each solid angle. Hence three addi- 

 tional conditions would be necessary to determine the value of every such angle, 

 and the problem would remain as indefinite as before. But if by any means 

 we can reduce the number of edges meeting in a point to four, only one con- 

 dition would be necessary to determine them all, and the problem would be 

 reduced to the consideration of one kind of bending only. 



This may be done by drawing the polyhedron in such a manner that the 

 planes of adjacent triangles coincide two and two, and form quadrilateral facets, 

 four of which meet in every solid angle. The bending of such a polyhedron 

 can take place only in one way, by the increase of the angles of two of the 

 edges which meet in a point, and the diminution of the angles of the other two. 



The condition of such a polyhedron being inscribed in any surface is then 

 found, and it is shewn that when two forms of the same surface are given, 

 a perfectly definite rule may be given by which two corresponding polyhedrons 

 of this kind may be inscribed, one in each surface. 



Since the kind of bending completely defines the nature of the quadrilateral 

 polyhedron which must be described, the lines formed by the edges of the 

 quadrilateral may be taken as an indication of the kind of bending performed 

 on the surface. 



These lines are therefore defined as " Lines of Bending." 



When the lines of bending are given, the forms of the quadrilateral facets 

 are completely determined ; and if we know the angle which any two adjacent 

 facets make with one another, we may determine the angles of the three edges 

 which meet it at one of its extremities. From each of these we may find the 

 angles of three other edges, and so on, so that the form of the polyhedron 

 after bending will be completely determined when the angle of one edge is given. 

 The bending is thus made to depend on the change of one variable only. 



In this way the angle of any edge may be calculated from that of any 

 given edge ; but since this may be done in two different ways, by passing 

 along two different sets of edges, we must have the condition that these results 

 may be consistent with each other. This condition is satisfied by the method 

 of inscribing the polyhedron. Another condition will be necessary that the 

 change of the angle of any edge due to a small change of the given angle, 

 produced by bending, may be the same by both calculations. This is the con- 

 dition of ' Instantaneous Lines of Bending." That this condition may continue 



