52 SCIENTIFIC APPARATUS. 



The importance of this surface in the undulatory theory of 

 Light forms its principal claim to attention. But, even apart from 

 any physical interpretation, its geometrical properties entitle it to 

 a place in a collection of geometrical models. It furnishes us 

 with an instance of a closed surface of two sheets, one of them 

 lying inside in the other ; it is an apsidal surface, and its reciprocal 

 surface is a surface of the same nature as itself; finally, it offers 

 typical examples of the singularity termed a conical node : and of 

 the correlative singularity of a tangent plane touching a surface 

 along a conic section (in the case of the wave surface the conic 

 section is a circle). The geometrical study of these singularities 

 led Sir William Rowan Hamilton to his celebrated discovery of 

 the optical phenomena of external and internal conical refraction. 



(c) The Surface of Steiner. 



Professor Cayley exhibits a rough model of this surface, which 

 has attracted considerable attention among mathematicians, from 

 its being the polar reciprocal of the cubic surface with four 

 conical nodes, and from its having the property that every .one of 

 its tangent planes cuts it in two conic sections. 



(d) The Amphigenous Surface of Professor Sylvester. 



This surface is of great importance in the theory of equations of 

 the fifth order. A model of it has been prepared by Professor 

 Henrici. 



(e) Surfaces of constant curvature. 



The total curvature of a surface at any point is the product of 

 its two principal curvatures at that point ; and the total curvature 

 is positive or negative, according as the two principal curvatures 

 are in the same direction or in opposite directions. It is an im- 

 portant geometrical theorem, that if two inextensible and flexible 

 surfaces have at corresponding points the same total curvature, 

 either of them can be " developed " upon the other without tearing 

 or crumpling. Thus every surface of constant positive curvature 

 can be developed upon a sphere. Surfaces of constant negative 

 curvature cannot, of course, be developed upon a sphere ; but 



