268 • G. H. KNIBBS. 



Dimensions (1) (2) (3) (4) 



Axial motions x x, y x, y, z x, y, z, w 



Rotations in the planes xy xy, xz xy, xz, xw 



yz yz, zw 



y...(8) 



Total Jn(n+1) = 1,3, 6 , 10 / 



that is to say the number of degrees of freedom of movement for 

 w-dimensional space is evidently Jn( + 1), or, what is the same 

 thing, each dimension increases the number of degrees of freedom 

 by the number which expresses the dimension itself. To consider 

 the point as the geometrical entity of dimension 0, is obviously 

 consistent with this formula. The matter of this special form of 

 generation, however, calls for no comment since it differs from 

 that previously considered, only in regard to the figure of the 

 element, and to the mere characteristics of the generative scheme, 

 and not essentially. 



1 2. Finitely and absolutely homaloidal 2 -dimensional space. — 

 Consider the expression 



xy = a (9) 



in which a may have any positive or negative value from — oo n to 

 + oo 11 . If in (9) xy denote a purely abstract product, and a an 

 abstract number also, the equation implies a purely numerical 

 relationship, between the three quantities, and one that, per se, is 

 independent therefore of all questions as to a possible geometrical 

 interpretation or significance. As such, that is as a purely 

 numerical relationship, we notice that the scale of either x or y 

 can be increased m times, or both can be equally increased Vm 

 times, by multiplying a by m, or both can be increased m times 

 by multiplying by m 2 . Hence there is no essential difference, 

 excepting one of scale, in any uniform scheme of geometrical 

 interpretation of this expression, i.e. (9). For example, if the 

 values of a be 1 , 2 , ...0 U the reciprocal relations of x and y are 

 completely defined by — 



