PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 267 



meaning however xA x , y.1 Y , etc., are fully denned by writing them, 



X= \\x; Y= \\xy; Z = \\xyz;...N = \%oyzw...eto (6) 



that is to say, xyzw etc. is a mere scalar product, and the linear, 

 square, cubic, quartic, n ic value of X, Y, Z, W...N, depends wholly 

 upon the value of the unit quantities 1}, 1", etc. 1 



Thus it is quite immaterial whether the units parallel to the 

 axes x, y, z etc. are unequal or equal. If the former, and measured 

 by some common linear unit they are a, b, c, d etc., and if also the 

 axes instead of being orthogonal differ from \k by the amounts 

 </>, x, i/s w, etc., then in orthogonal ^-dimensional units, coinciding 

 with that by which a, 6, etc. are measured, 



1* = {abed. . . )(cos <f> cos x cos x// cos to . . . ) (6a) 



Hence obviously the numerical representation of space, and the 

 spatial interpretation of numerical values ought logically to be 

 kept distinct. 



We observe also that the ordinary loose way of writing mO = 0, 

 mj oo = 0, 0/0 etc., makes operation upon such quantities subject 

 to uncertainty, and further that ooO is unity, 2 only for equivalent 

 dimensions, since 



0»x oo = 0r 1 ; 0»x ao ul = 0r m (7) 



No uncertainty can arise so long as scalar and vector parts of each 

 operation, are carefully distinguished, and the real magnitudes 

 of zero and infinities are retained. 



11. Rotational generation. — Generation by rotations, or by 

 rotation combined with axial motion, does not essentially differ 

 from the latter alone, and is of course continuous, n-dimensional 

 figures will of course, be generated by motion of an (n — ^-dimen- 

 sional generatrix whenever it moves into the ?*-th dimension. The 

 total number of ways in which generative motion can take place, 

 may be enumerated as follows, viz.: — 



1 In practice the index and suffix are of course not required, because 

 the number of axes x, y, z etc. denotes both at the same time. Because 

 in pure numbers 1 x 1 x . . • 1 to n figures can be represented only by 1, we 

 are apt to forget that l 1 and l n may be very different quantities. 



2 See Chrystal, Text Book of Algebra, n., pp. 66 - 96, 1889. 



