PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 285 



'Batoo 



Fig.15 W 



hence for 6 = 0° or 180°, / is infinite; and when </> + 6 = 1 80° or , 

 /is zero; the infinity and zero being of the same order. Again 

 if/ be zero, that is if S be infinitesimally close to the line RB, / 

 becomes zero, for all finite values of <f> and 6 excluding zero; con- 

 versely if g be zero, /being finite, that is if S be infinitesimally 

 close to the line ab, then I is infinite for all finite values of <f> and 

 excluding zero. As soon as different orders of infinitesimals 

 are considered, it will be seen that the infinities must be of the 

 same order ; and not only so, but by operating on the ratio f/g, so 

 as to make it an infinity or zero of any required order, we can by 

 similar or the converse treatment of 0, obtain infinities or zeros 

 either of a still higher order, or on the other hand may maintain 

 the intensity constant. This would not be true if the lines were 

 infinitesimally curved, i.e. if p at oo were really p' at oo, see 

 Fig. 14. 



18. Space of non-uniform intensity. — The idea of spatial varia- 

 tion of intensity, which has been illustrated in the preceding 

 section, for a line, through linear projection, may be applied to 



