447] ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 
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x : y = X' : Y', where X', Y' are linear functions of (xy'). Thus in the present case, 
instead of an infinity of transformations for different values of n, n!, we have only the 
well-known homographie transformation wherein n=n'—l. 
8. In the discussion of the foregoing cases of the transformation between two 
planes and two spaces, it was tacitly assumed that n was greater than 1, and the 
transformations considered were thus different from the homographie transformation ; 
but it is hardly necessary to remark that the homographie transformation applies to 
these cases also ; viz., for two planes we may have x' : y' : z' = X : F : Z, and 
x : y : z = X' : Y' : Z', where (X, Y, Z), (XY', Z') are linear functions of the two 
sets of coordinates respectively ; and similarly for two spaces ad : y[ : z' : w' = X : F : Z : W 
and x : y : z : w = X' : Y' : Z' : W', where (X, F, Z, W), (XY r , Z', W') are linear 
functions of the two sets of coordinates respectively. We may, if we please, separate 
off the homographie transformation (as between two lines, planes, and spaces respectively), 
and restrict the notion of the rational transformation to the higher or non-linear trans 
formations ; in this point of view, the case of two lines would not be considered at 
all, but the theory of the rational transformation would begin with the case of the 
two planes. Such severance of the theory is, however, somewhat arbitrary ; and more 
over the homographie transformation between two lines (being, as mentioned, the only 
rational transformation) is analogous not only to the homographie transformation between 
two planes, and to the homographie transformation between two spaces, but it is also 
analogous to the lineo-linear (or quadric) transformation between two planes, and to the 
lineo-linear (which is a cubic) transformation between two spaces. 
9. For the sake of bringing out this analogy, I shall consider in some detail the 
homographie transformation between two lines ; but as regards the homographie trans 
formations between two planes and between two spaces respectively (although there is 
room for a like discussion) the theories may be considered as substantially known, and 
I do not propose to go into them. 
The Homographie Transformation between Two Lines. 
10. By what precedes, it appears that we have x : y' = X : Y, where (X, Y) are 
linear functions of (x, y); and conversely, x : y — X' : Y', where X', Y' are linear 
functions of (x\ y); or what is the same thing, the relation is expressed by a single 
equation 
(ax + by) x' + (cx + dy) y' = 0 ; 
or, as this may conveniently be written, 
or, when the expression of the actual values of the coefficients is unnecessary, 
(*$>, y) y') = °- 
We thus see that the rational transformation between two lines is in fact the homo 
graphic transformation; and also that it is the lineo-linear transformation. 
C. VII. 
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