286 G. H. KNIBBS. 



space of any dimensions. 1 The variation in the case illustrated 

 ranges between zero and infinity in a simple manner, which may 

 be defined by the equation 



P = k sin (<£ + 0) cosec 6 (23a). 



The projections of curves however will give lines of more complex 

 variations of intensity. In the generation of geometrical figures, 

 the generating elements may be assumed to follow any particular 

 law of intensity in moving along, or in rotating about any axis, 

 and consequently the generated space will not be homogeneous or 

 isotropic, 2 though it may be homaloidal. Geometries of such space 

 are possible; the intensities may preferably be supposed to vary 

 with absolute continuity, 



19. Complex Space. — Not only may space be assumed to be 

 non-uniform in intensity, so that the spatial distribution of intensity 

 shall follow any assigned law, but it may be supposed further to 

 conform to different laws in respect of different properties. Such 

 space may be called complex seolotropic space. Geometries which 

 attempt to take simultaneous account of a series of laws of 

 intensity will necessarily be extremely difficult. 



20. Space of positive and negative curvature. — As already stated 

 in supposed more general conceptions of "space" euclidean is pro- 

 posed to be treated as a merely special or limiting case. "Space" 

 in which two "straight" lines 3 can intersect only once, is known 

 as hyperbolic space, and is infinite in some higher sense than 



1 So far as I am aware, the formal recognition of intensity as applicable 

 to any existing thing is due to Kant. See op. cit. "Anticipation en der 

 Wahrnehmung." " Das Princip derselben ist : In alien Erscheinungen 

 hat das Reale, was ein Gegenstand der Empfindung ist, intensive Grosse, 

 d. i einen Grad. ... So hat demnach jede Empfindung, mithin auch 

 jede Realitat in der Erscheinung, so klein sie auch sein mag, einen Grad, 

 . . . die noch immer vermindert werden kann, und zwischen Eealitat 

 und Negation ist ein kontinuirlicher Zusammenhang moglicher Realitaten 

 . . . ." Elementarlehre, Buch n. Haupst 2, Abschn. 3 ii. 



8 Such conceptions are frequently realisable in physics, as for example 

 an electric or magnetic field, variations in the density of bodies, non- 

 uniform distributions of heat or other forms of energy, etc. These may 

 all be regarded as space of non-uniform intensity, and a suitable geometry 

 could be developed for each type of variation. 



3 Defined as the shortest lines between any two points, and supposed 

 to be like the similar lines on three-dimensional surfaces. 



