GEOMETRICAL INSTRUMENTS AND MODELS. 47 



order (the cone, the cylinder, the ellipsoid, the hyperboloids, and 

 both the paraboloids) is also exhibited by Professor Henrici, of 

 University College, and by Professor Brill, of Munich. These 

 models exhibit very clearly the circular sections of the various 

 surfaces, having, indeed, been constructed by means of them ; with 

 the exception of the hyperbolic paraboloid, in which (as well as 

 in the hyperbolic and parabolic cylinders) such sections (strictly 

 speaking) do not exist. 



(2.) Models of surfaces of the third order. 



Nothing like a complete series of models of surfaces of the 

 third order has as yet been attempted; and indeed it may be 

 said that our knowledge of these surfaces is still too imperfect to 

 justify such an attempt. Nevertheless, models of certain of these 

 surfaces have been, made, and we may mention a few important 

 properties which they serve to illustrate. 



(a.) As we have already said, the two curvatures of an ellipsoid 

 at any point upon the surface are turned the same way, whereas 

 the two curvatures of an hyperboloid are everywhere turned 

 opposite ways. But, generally speaking, a curve surface consists 

 of two regions, on one of which its two curvatures are turned the 

 same way, and on the other opposite ways. These two regions 

 are separated by a bounding line, technically called the parabolic 

 curve. Surfaces of the third order offer typical examples of these 

 general geometrical facts. But an example, though of a less perfect 

 kind, is afforded by the figure of a ring (such as an anchor ring, or 

 a wedding-ring); the parabolic curve here consisting of either of 

 the two circles on which the ring would rest if placed lying on a 

 horizontal plane. 



(b.) Wherever the two curvatures of a surface are opposite to 

 one another, there always exist upon the surface two sets of lines 

 along which the surface is inflected ; i.e. at any point of the surface 

 these two lines separate the directions in which the curvature is 

 turned one way from those directions in which it is turned the 

 other way. On the hyperboloid these " curves of principal tan- 



