456 Mb MAXWELL, ON THE 



intersection must be parallel to two conjugate diameters of the conic of contact at that point 

 in order that such a polyhedron may be inscribed. 



Systems of curves intersecting in this manner will be referred to as "conjugate systems. 1 ' 



5. On the elementary conditions of the applicability of two surfaces. 



It is evident, that if one surface is capable of being applied to another by bending, every 

 point, line, or angle in the first has its corresponding point, line, or angle in the second. 



If the transformation of the surface be effected without the extension or contraction of any 

 part, no line drawn on the surface can experience any change in its length, and if this condi- 

 tion be fulfilled, there can be no extension or contraction. 



Therefore the condition of bending is, that if any line whatever be drawn on the first 

 surface, the corresponding curve on the second surface is equal to it in length. All other con- 

 ditions of bending may be deduced from this. 



6. If two curves on the first surface intersect, the corresponding curves on the second 

 surface intersect at the same angle. 



On the first surface draw any curve, so as to form a triangle with the curves already 

 drawn, and let the sides of this triangle be indefinitely diminished, by making the new curve 

 approach to the intersection of the former curves. Let the same thing be done on the second 

 surface. We shall then have two corresponding triangles whose sides are equal each to each, 

 by (5), and since their sides are indefinitely small, we may regard them as straight lines. 

 Therefore by Euclid I. 8, the angle of the first triangle formed by the intersection of the two 

 curves is equal to the corresponding angle of the second. 



7. At any given point of the first surface, two directions can be found, which are 

 conjugate to each other with respect to the conic of contact at that point, and continue 

 to be conjugate to each other when the first surface is transformed into the second. 



For let the first surface be transferred, without changing its form, to a position such 

 that the given point coincides with the corresponding point of the second surface, and the 

 normal to the first surface coincides with that of the 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. 



