TRANSFORMATION OF SURFACES BY BENDING. 97 



Two systems of curves being drawn on the first surface, the corresponding 

 systems may be drawn on the second surface. These systems being conjugate 

 to each other, fulfil the condition of Art. (4), and may therefore be made the 

 means of constructing a polyhedron with quadrilateral facets, by the bending of 

 which the transformation may be effected. 



These systems of curves will be referred to as the "first and second systems 

 of Lines of Bending." 



9. General considerations applicable to Lines of Bending. 



It has been shewn that when two forms of a surface are given, one of 

 which may be transformed into the other by bending, the nature of the lines 

 of bending is completely determined. Supposing the problem reduced to its 

 analytical expression, the equations of these curves would appear under the 

 form of double solutions of differential equations of the fir^t order and second 

 degree, each of which would involve one arbitrary quantity, by the variation of 

 which we should pass from one curve to another of the same system. 



Hence the position of any curve of either system depends on the value 

 assumed for the arbitrary constant ; to distinguish the systems, let us call one 

 the first system, and the other the second, and let all quantities relating to 

 the second system be denoted by accented letters. 



Let the arbitrary constants introduced by integration be u for the first 

 system, and u' for the second. 



Then the value of u will determine the position of a curve of the first 

 system, and that of u a curve of the second system, and therefore u and u' will 

 suffice to determine the point of intersection of these two curves. 



Hence we may conceive the position of any point on the surface to be 

 determined by the values of u and u for the curves of the two systems which 

 intersect at that point. 



By taking into account the equation to the surface, we may suppose x, y, 

 and z the co-ordinates of any point, to be determined as functions of the two 

 variables u and u. This being done, we shall have materials for calculating 

 everything connected with the surface, and its lines of bending. But before 

 entering on such calculations let us examine the principal properties of these lines 

 which we must take into account. 



Suppose a series of values to be given to u and u, and the corresponding 

 curves to be drawn on the surface. 



VOL. i. 13 



