448 Mr MAXWELL, ON THE 



£ constant. If these surfaces be bent so that = 0, the whole of the generating lines will lie 

 in one plane, and their ultimate intersections will form a plane curve. The surface is thus 

 reduced to one plane, and therefore belongs to the class usually described as "developable 

 surfaces." The form of a developable surface may be defined by means of the three quantities 

 9, a and d). The generating lines form by their ultimate intersection a curve of double 

 curvature to which they are all tangents. This curve has been called the cuspidal edge. 



The length of this curve is represented by <r, its absolute curvature at any point by — , 



• , d 

 and its torsion at the same point by — - . 



da 



When the surface is developed, the cuspidal edge becomes a plane curve, and every part 

 of the surface coincides with the plane. But it does not follow that every part of the plane 

 is capable of being bent into the original form of the surface. This may be easily seen by 

 considering the surface when the position of the cuspidal edge nearly coincides with the plane 

 curve but is not confounded with it. It is evident that if from any point in space a tangent 

 can be drawn to the cuspidal edge, a sheet of the surface passes through that point. Hence 

 the number of sheets which pass through one point is the same as the number of tangents to 

 the cuspidal edge which pass through that point ; and since the same is true in the limit, the 

 number of sheets which coincide at any point of the plane is the same as the number of 

 tangents which can be drawn from that point to the plane curve. In constructing a de- 

 velopable surface of paper, we must remove those parts of the sheet from which no real 

 tangents can be drawn, and provide additional sheets where more than one tangent can be 

 drawn. 



In the case of developable surfaces we see the importance of attending to the position of 

 the lines of bending; for though all developable surfaces may be produced from the same 

 plane surface, their distinguishing properties depend on the form of the plane curve which 

 determines the lines of bending. 



II. 



On the Bending of Surfaces of Revolution. 



In the cases previously considered, the bending in one part of the surface may take place 

 independently of that in any other part. In the case now before us the bending must be 

 simultaneous over the whole surface, and its nature must be investigated by a different method. 



The position of any point P on a surface of revolution 

 may be determined by the distance PV from the vertex, 

 measured along a generating line, and the angle AVO which 

 the plane of the generating line makes with a fixed plane 

 through the axis. Let PV=s and AVO = 9. Let r 

 be the distance (Pp) of P from the axis, r will be a function 

 of « depending on the form of the generating curve. 



