PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 305 



figures may be involved. In Fig. 27 the effect of projection on 

 intensity is more fully illustrated. Suppose OKL...T to be a 

 curve of uniform intensity radially projected from the point C on 

 to the curves 0, 1...4. The simplest relationship will be on the 

 straight line (0); on the circular curve (1) the relation will be 

 more complex; and, on the curve of changing curvature, viz. (2) 

 still more so. Although at the points k, 1, 1', r, q in (2) the 

 intensity is infinite, 1 it is finite for all finite distances however 

 small. At a point like m it is zero, but at one like k', where the 

 line CK is tangential to both curves, it depends upon the ratio of 

 the differentials of the curves, CK being adopted as axis. The 

 mean intensity of the stretch ps is the total length PQRS 

 divided by ps; and similarly throughout. A moving point, in gener- 

 ating any line, may be assumed to vary its intensity in any given 

 way: so also in regard to a line generating a surface, a surface 

 generating a solid, a solid generating a fourth dimensional quantum 

 and so on. The possible schemes of varying intensity are obviously 

 illimitable, consequently the complexity of specialised regions of 

 space may be varied in an infinite number of ways. It is evident 

 from this point of view that the so-called elliptic space, and 

 hyperbolic space are simply some of the most simple forms of 

 specialised regions of space. 2 



In generating an seolotropic or heterogeneous spatial region, 

 say of spherical (or ellipsoidal) form, its heterogeneity or seolotropy 

 may vary periodically on non-periodically along the radii, or in 

 the parallel planes dividing it into circular (or elliptic) sections; 

 and the variation of the intensity on the radius in the xy plane, 

 may be periodic as it rotates in that plane. Thus, as this plane 

 moves along the z axis, the points of maxima and minima may be 

 rotated, and the period so varied, that if the seolotropic variation 

 be continuous, the lines of maximum intensity, would form a spiral 

 of varying intensity. Fig. 28 will afford an illustration, the 



1 The radial lines being tangential to the curve at K, L, E and Q. 



2 A magnetic or electric field may be instanced as an elementary 

 example of specialised space. 



T— Dec. 4, 1901. 



