447] 
189 
447. 
ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 
[From the Proceedings of the London Mathematical Society, vol. ill. (1869—1871), 
pp. 127—180. Account of the Paper given at the Meeting 11 March 1869.] 
Two figures are rationally transformable each into the other (or, say, there is a 
rational transformation between the two figures) when to a variable point of each of 
them there corresponds a single variable point of the other. The figures may be 
either loci in a space, or locus in quo of any number of dimensions; or they may 
be such spaces themselves. Thus the figures may be each a line (or space of one 
dimension), each a plane (or space of two dimensions), or each a space of three 
dimensions; these last are the cases intended to be considered in the present Memoir, 
which is accordingly entitled, “ On the Rational Transformation between Two Spaces.'’ 
I observe in explanation (to fix the ideas, attending to the case of two planes), that 
any rational transformation between two planes gives rise to a rational transformation 
between curves in these planes respectively (one of these curves being any curve what 
ever): but non constat, and it is not in fact the case, that every rational transformation 
between two plane curves thus arises out of a rational transformation between two 
planes. The problem of the rational transformation between two planes (or generally 
between two spaces) is thus a distinct problem from that of the rational transformation 
between two plane curves (or loci in the two spaces respectively). 
I consider in the Memoir, (1) the rational transformation between two lines; 
this is simply the homographic transformation: (2) the rational transformation between 
two planes; and here there is little added to what has been done by Prof. Cremona 
in his memoirs, “ Sulle Trasformazioni Geometriche delle Figure Piane,” (Mem. di 
Bologna, t. ii., 1863, and t. v., 1865; see also “ On the Geometrical Transformation 
of Plane Curves,” Br.tish Assoc. Report, 1864) : (3) the rational transformation between 
two spaces; in regard hereto I examine the general theory, but attend mainly to 
what I call the lineo-linear transformation; viz., it is assumed that the coordinates