304 G. H. KNIBBS. 



a function of the direction of any point frcm the fixed point or 

 modular focus, this may be called complex modular generation. 

 Modularly generated surfaces are of the second degree if the ratio 

 be constant. When the square of the distance from the focal 

 point 1 to the point on the surface bears a constant ratio 2 to the 

 rectangle whose sides are the perpendiculars to two fixed planes, 3 

 the surface is said to be umbilically generated, and is a surface of 

 the second degree, or a conicoid. 4 



As in modular, so in umbilical generation, the modulus may be 

 continuously varied, in which case it may be called complex 

 modular generation. 



It is obvious that by variations of the moduli, any system of 

 points generated may lie on a surface of any degree of complexity. 

 These methods of generating geometrical figures give point-systems, 

 rather than continuous surfaces. 



30. Generation of figures of non-uniform intensity. — Figs. 13, 

 14, and 25 afford examples of lines of non-uniform intensity, the 

 law of variation of the intensity being simple in each case. In 

 generating any geometrical figure, the generatrix, conceived as 

 imparting its own characteristics to the space through which it 

 passes, may itself be of non-uniform intensity; its intensity may 

 vary during the generative motion in any specified way; a 

 geometrical figure may be obtained not directly from the genera- 

 tive scheme, but from the projection of the generated figure; or, 

 it may be, from any combination of projections. In these and 

 many other ways, the heterogeneity and seolotropy 5 of geometrical 



1 The umbilical, instead of modular, focus. 8 The umbilical modulus- 



3 The planes are called directing planes, and their intersection the 

 directrix. 



4 The surfaces that can be generated (with real focus and directrix) 

 are the ellipsoid, hyperboloid of two sheets, elliptic paraboloid, a cone 

 point. The hyperboloid of revolution of two sheets and the prolate 

 spheroid can be umbilically generated. 



6 It is generally convenient to restrict the antithetical words "homo- 

 geneous" and "heterogeneous" to the implication of mere differences of 

 density; and "isotropic" and " se )lotropic " to differences in the pro- 

 perties of a body, which may nevertheless be of uniform density. As the 

 idea of intensity is not by any means synonymous with density, geolotropic 

 will perhaps convey the more general meaning. 



