PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 263 



No quantity can be generated in the (ra + l)th dimension by 

 summational operations on quantities of the mth dimension. 



It is now evident that with infinitesimals of the nth dimension, 

 denned by xyzw-etc> the summational operations may be wholly 

 restricted to any lower dimension, or on the other hand, may 

 develope in any order, through the series of axial directions up to 

 and inclusive of the n-th dimension itself, but not beyond it. 

 Remembering that the several axes may be interchanged, the 

 following series of symbols will sufficiently indicate the successive 

 developments by means of infinite summation of the 1st, 2nd, and 

 3rd and 4th orders. We take as generatrix a 4th dimensional 

 infinitesimal, the various developmental forms of which are shewn 

 in the first line, and the sum to oo, oo 2 etc. in the successive 

 lines downwards. 



Magnif udes of Generated Quanta. 



■(*) 



It will be noticed that numerically 0* == 0.1 n_1 or more generally 

 k = k l u-k . Let the generated quanta be denoted by 



a e i m 



b f j n 



c g k o 



d h I p 



they may be described as follows : — Linear elements, (a) a linear 

 unit; (b) and infinite line, (c) an infinite line of the second order; 1 

 (d) and infinite line of the third order. Surface elements, (e) a 

 linear surface-zero, that is such as zero as would be represented 

 by the parallelogram l x y ; (f) a parallelogrammic surface unit; 2 



1 That is a line such that its unit is an infinity of the first order, and 

 its zero-element a unit of the first order. 



2 If the axes are orthogonal, substitute "rectangular" for "parallelo- 

 grammic": if the units are also equal substitute "square." 



Order of 

 Summation 



(1) 



(2) 



(3) (4) 





1 



o x o yzw = 



o xy OL = 



3 1 = 4 



u xyz u w w xyzw 



etc. 



OO 



1 3 = 



1 2 = 



2 1 = 3 



etc. 



oo 2 



OOxOyzw = 



l 2 2 = 



1 xy u zw 



1 1 = 2 



v xyz v w u xyzw 



etc. 



oo 3 



oo x 3 zw = 



O°xy0L = 



p Ql Ql 



1 xyz v/ w w xyzw 



etc. 



oc 4 



<*>*o 3 yzw = 



^xyOL = 



OO 1 = l 4 



etc. 



etc. 



etc. 



etc. 



etc. etc. 





