PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 299 



absolutely so. The intensity of the lines OP, OQ Fig. 24 being 

 uniform, and OX in Fig. 25, the intensity of the curve QO'S in 

 the latter figure is not uniform, that is to say its linear value does 

 not coincide with that of corresponding points in QO'S Fig. 24. 1 

 Hence the integral of the curve O'S, for its length as in Fig. 25, is 



s=fv{l+(dy/dxf}dx.... (27) 



while the real length of the line as in Fig. 24 is: — 



S=fl<s{l+(dyldx)*]dx (27a) 



I being of course a function of x, as indicated in the preceding 

 footnote. 2 It is evident therefore that a complete theory of 

 relativity must include relativity of intensity as well as mere 

 relativity of position, (See also § §, 16, 17, 18), and specialisations 

 of space are not inversely comparable unless both intensity and 

 position are taken into account. This however implies that the 

 results are general only when the space is a definite specialisation 

 of an isotropic, homogeneous homaloid. The cases cited, illustra- 

 tive of a relativity of position and intensity, indicate that its 

 complexity has no limit. 



28. Complex generation of geometrical figures of uniform 

 intensity. — In the preceding sections, treating of the generation 

 by summational, fluxional, or rotational operations, 3 no account 

 was taken of more complex processes. If the elements from which 

 a circle of radius r is built up (summationally) be of intensity a 

 in one direction, and (3 at right angles thereto, its intensity-area 

 is 7raftr 2 ; that is it is equivalent to the ellipse with semi-axes ar, 

 fir; and if it be supposed to have been compressed and to expand 

 till the intensity was uniform, it would be actually an ellipse with 

 those axes. 



1 Let 8$ denote the length of any small element of the line 0'S,andSs 

 the length of the corresponding element in the curve O'S (Fig. 25). Then 

 the intensity I, see (23) § 17, is 



I = 88j8s = /x/V(cos 4 xjp + fM sin 3 x/p) 

 /x denoting l + b/p. 



2 The lengths corresponding to points 1, 2, 3 etc., on the line O'S Fig. 

 24, are marked by the points i., ii., iii. etc., on Fig. 25. The distance 

 from the corresponding 'arabic' numerals shews the difference. 



3 §§ 2-6, and 11. 



