96 



TRANSFORMATION OF SURFACES BY BENDING. 



second at the same point. Then let the first surface be turned about the normal 

 as an axis till the tangent of any line through the point coincides with the 

 tangent of the corresponding line in the second surface. 



Then by (6) any pair of corresponding lines passing through the point will 

 have a common tangent, and will therefore coincide in direction at that point. 



If we now draw the conies of contact belonging to each surface we shall 

 have two conies with the same centre, and the problem is to determine a pair 

 of conjugate diameters of the first which coincide with a pair of conjugate 

 diameters of the second. The analytical solution gives two directions, real, 

 coincident, or impossible, for the diameters required. 



In our investigations we can be concerned only with the case in which these 

 directions are real. 



When the conies intersect in four points, P, Q, R, S, PQRS is a parallelo- 

 gram inscribed in both coriics, 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 con- 

 centric 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 conju- 

 gate 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. 



