46 SCIENTIFIC APPARATUS. 



right angles to one another, along one of which the curvature is 

 the greatest, and along the other the least. At special points, 

 called umbilics, the greatest and least curvatures (and therefore 

 all the curvatures) are equal to one another. The sphere has the 

 peculiarity that every point on it is an umbilic ; on the sphere, 

 therefore, there are no directions of greatest and least curvature ; 

 but on every other surface two series of curves can be drawn, cutting 

 one another at right angles, and indicating for each point of the 

 surface the directions of the greatest and least curvature. These 

 lines are called the lines of curvature ; they are, it may be said, 

 detected by the eye itself on any surface. The ellipsoid has four 

 umbilics ; if a thread, attached to any two of these, be strained 

 along the surface by a moving pencil, the pencil will describe a 

 line of curvature, so that when the ellipsoid has once been 

 modelled, and its umbilics determined, it is easy to draw its lines 

 of curvature with sufficient approximation. It was suggested long 

 ago by Monge that a semi-ellipsoidal vault would form the most 

 appropriate covering for an oval room, that the natural lines of 

 the vaulting would be the lines of curvature, and that the umbilics 

 would be the proper centres of illumination. 



Every surface of the second order is either umbilical or rectili- 

 neal; i.e., it either possesses umbilics, or it is capable of being 

 generated by the motion of a straight line : it cannot unite both 

 properties,. The ellipsoid, as we have just seen, is umbilical ; the 

 hyperboloid of one sheet is rectilineal ; and the two systems of 

 straight lines which lie upon this surface are very conspicuous in 

 any model of it. It will be seen that one line of each system 

 passes through each point on the surface ; but that no two lines 

 of the same system ever meet one another. The hyperbolic para- 

 boloid, which may be regarded as a variety of the hyperboloid of 

 one sheet, is characterized by similar properties : the Exhibition 

 includes a beautiful series of models of these two surfaces made 

 by M. Fabre de Legrange, in 1872, for the ^South Kensington 

 Museum. A series of cardboard models of surfaces of the second 



