TRANSFORMATION OF SURFACES BY BENDING. 457 



When the conies intersect in four points, P, Q, R, S, PQRS 

 is a parallelogram inscribed in both conies, and the axes CA, CB, 

 parallel to the sides, are conjugate in both conies. 



If the conies do not intersect, describe, through any point 

 P of the second conic, a conic similar to and concentric 

 with the first. If the conies intersect in four points, we must 

 proceed as before ; if they touch in two points, the diameter 

 through those points and its conjugate must be taken. If they 

 intersect in two points only, then the problem is impossible; and 

 if they coincide altogether, the conies are similar and similarly 

 situated, and the problem is indeterminate. 



8. Two surfaces being given as before, one pair of conjugate systems of curves may 

 be drawn on the first surface, which shall correspond to a pair of conjugate systems on 

 the second surface. 



By article (7) we may find at every point of the first surface two directions conjugate 

 to one another, corresponding to two conjugate directions on the second surface. These 

 directions indicate the directions of the two systems of curves which pass through that point. 



Knowing the direction which every curve of each system must have at every point of 

 its course, the systems of curves may be either drawn by some direct geometrical method, 

 or constructed from their equations, which may be found by solving their differential 

 equations. 



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 first 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. 



59 — 2 



