450 Mr MAXWELL, ON THE 



equations might be found connecting the angles of the different edges which meet in each solid 

 angle of the polyhedron. It will be shown that the conditions thus obtained would be equi- 

 valent to three equations between the six angles of the edges belonging to each solid angle. 

 Hence three additional conditions would be necessary to determine the value of every such 

 angle, and the problem would remain as indefinite as before. But if by anv means we can 

 reduce the number of edges meeting in a point to four, only one condition 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 

 shown 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 condition of " Instantaneous 

 Lines of Bending." That this condition may continue to be satisfied during the whole pro- 

 cess we must have another, which is the condition for " Permanent Lines of Bending." 



The use of these lines of bending in simplifying the theory of surfaces is the only part of 

 the present method which is new, although the investigations connected with them naturally led 

 to the employment of other methods which had been used by those who have already treated of 

 this subject. A statement of the principal methods and results of these mathematicians will 

 save repetition, and will indicate the different points of view under which the subject may pre- 

 sent itself. 



The first and most complete memoir on the subject is that of M. Gauss, already 

 referred to. 



