V x -V=iirax O p ; - irT ; - — ; +logr; + r; -r 2 ; 



PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 307 



The part to be added to the volume to correct it for intensity is 

 therefore for n = - 3, and generally 



F, - V= 4:ira log r, and iira (32) 



respectively. The values of this are 1 



Values of 

 n = -oo -5 -4 -3 -2-10+OO 



1 1 , 1 1 2 1 3 



2"^ ; ~v> + ogr; +r; 2 ; 3 ' 



When n is between - 4 and - oo, the intensity at the point O is 

 infinite : on the other hand it may be made zero at any distance 

 from that we please, by taking a negative and suitably choosing 

 a and n. Beyond this distance the intensity will be negative. 

 Negative intensity may be defined as any affection of a spatial unit, 

 such that if it be combined with an equal but opposite affection of 

 an equal unit, the result will be null? Zero-intensity implies the 

 non-existence of the affection. 



32. Conversion of ceolotropic into isotropic space. — Let OX, 

 Fig. 29, denote the axis of a cone POP' in an seolotropic homaloid 

 of three dimensions : and first let it be supposed that the density 

 in a line, x to y, perpendicular to the axis, is uniform but increases 

 as x, the distance from O, increases. Suppose the cone to expand 

 parallel to XP only, till the density was uniform, the form would 

 be similar to HOH'. What were originally straight lines would 

 be bent outwards and we should have all the characteristics of 

 hyperbolic, developed from parabolic space. 3 Conversely, suppose 

 the density to diminish as we move along OX, and the cone to be 

 similarly compressed till of uniform density, we should have as 



1 If n = and a = 1, which, see (29), means that we assume the intensity 

 to be zero throughout, the result for the intensity-volume, is of course 

 zero also. 



2 Let V denote the potential at a point P of a sphere of density 28 in a 

 region of space of density 8. (i.e. the sphere has an excess of density d 

 over that of the space in which it lies) and let V be the potential of the 

 void space, when the sphere is removed (i.e. when the defect of density is 

 6). Then V' - -V, V and V maybe regarded as equal and opposite 

 intensities, i.e. as + I and —I : so also in this case + 8 and - 8 may be 

 regarded as equal and opposite the affections of space. 



3 See the right hand side of Fig. 20. 



