TRANSFORMATION OF SURFACES BY BENDING. 



455 



surface, and when the size of the triangle is indefinitely diminished, it will approximate to the 

 form of a conic section. 



For we may suppose a surface of the second order constructed so as to have a contact of 

 the second order with the given surface at a point within the angular points of the triangle. 

 The curve of intersection with this surface will be the conic section to which the other curve 

 of intersection approaches. This curve will be henceforth called the " Conic of Contact, 1 ' 

 for want of a better name. 



To find the radius of curvature of a normal section of the 

 surface. 



Let ARa be the conic of contact, C its centre, and CP per- 

 pendicular to its plane. rPR a normal section, and its centre 

 of curvature, then 



™ 1 PR* 

 2 CP 



1 CR* 

 = — -— in the limit, when CR and PR coincide, 



1 rR? 

 ~8 CP" 

 or calling CP the " sagitta," we have this theorem : 



" The radius of curvature of a normal section is equal to the square of the corresponding 

 diameter of the conic of contact divided by eight times the sagitta." 



4. To inscribe a polyhedron in a given surface, all ivhose sides shall be plane quadri- 

 laterals, and all whose solid angles shall be tetrahedral. 



Suppose the three systems of curves drawn as described in sect, (l), then each of the 

 quadrilaterals formed by the intersection of the first and second systems is divided into two 

 triangles by the third system. If the planes of these two triangles coincide, they form a 

 plane quadrilateral, and if every such pair of triangles coincide, the polyhedron will satisfy the 

 required condition. 



Let abc be one of these triangles, and acd the 

 other, which is to be in the same plane with abc. Then 

 if the plane of abc be produced to meet the surface in 

 the conic of contact, the curve will pass through abc 

 and d. Hence abed must be a quadrilateral inscribed 

 in the conic of contact. 



But since ab and dc belong to the same system of curves, they will be ultimately paral- 

 lel when the size of the facets is diminished, and for a similar reason, ad and be will be 

 ultimately parallel. Hence abed will become a parallelogram, but the sides of a parallelo- 

 gram inscribed in a conic are parallel to conjugate diameters. 



Therefore the directions of two curves of the first and second system at their point of 

 Vol. IX. Part IV. 59 



