'.'I 



TRANSFORMATION OF SURFACES BY BKMHN'. 



/ //in/ (hi- ifidt'uft <)/' atnftture of a normal section 

 <>f the surface. 



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

 < '/' perpendicular to its plane. rPR a normal section, and 

 its centre of curvature, then 



PIP 



PO - 

 ~ 



Clf 

 = \-fTp in the limit, when CR and P7? coincide. 



r/? 

 -t 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 wJiose side* */<// be 

 /ilnnc (quadrilaterals, and all whose solid angles shall be tctmln-ilral. 



Suppose the three systems of curves drawn as described in sect. (1), 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 quad- 

 rilateral inscribed in the conic of contact. 



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

 ultimately parallel 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 paral- 

 lelogram, but the sides of a parallelogram inscribed in a conic are parallel to 

 conjugate diameters. 



