ME. A. CAYLEY’S SIXTH MEMOIE UPON QUANTICS. 
77 
have a line and a point, and if lines are drawn joining the point with the several points 
of the line, these lines constitute a pencil, and the points of the line constitute a range, 
and such pencil and range are homographically related. 
194. The theory of homography in geometry of two dimensions may be made to 
depend upon the corresponding theory in geometry of one dimension, or what is the 
same thing, upon the homography of ranges or pencils. For consider two figures exist- 
ing iti distinct planes or spaces of two dimensions, any four points (not in a line) in the 
second figure may correspond to any four points (not in a line) in the first figure ; and 
when this is so, we may, by the process about to be explained, given any other point of 
the first figure, construct the corresponding point of the second figure ; and the two 
figures are then, by definition, homographically related. Suppose that the points 
A', B', C', D' of the second figure correspond respectively to the points A, B, C, D of 
the first figure, and let E be any other point of the first figm’e ; suppose that E' is the 
corresponding point of the second figure ; the pencils AB, AC, AD, AE and A'B', A'C', 
A'D', A'E' should be homographic to each other, that is, E' must lie on a given line 
through A' ; and in like manner the pencils BA, BC, BD, BE and B'A', B'C', B'D', B'E' 
should be homographic to each other, that is, E' must lie on a given line through B'. 
And then, as a theorem, CA, CB, CD, CE and C'A', C'B', C'D', C'E', or DA, DB, DC, DE 
and D'A', D'B', D'C', D'E' will be homographic pencils, that is, the construction will be 
a determinate one whichever two of the four points are selected for the points A and B. 
The foregoing construction leads to an analytical relation, which I think constitutes a 
better foundation of the theory. Consider the first plane as the locus m quo of the 
coordinates {cc, y, z), and the second plane as the locus in quo of the coordinates (X, Y, Z), 
these two coordinate systems being absolutely independent of each other. Consider any 
point of the first plane and a corresponding point of the second plane such that its 
coordinates (X, Y, Z) are given linear functions of the coordinates {x, y, z) of the point 
in the first plane. Any figure whatever in the first plane gives rise to a corresponding 
figm’e in the second plane, and the two figures are said to be homographic to each other. 
To a point of the first figure there corresponds in the second figure a point, to a line a 
line, to a range of points or pencil of lines, a homographic range of points or pencil of 
lines ; the line or point which is the locus in quo of the range or pencil in the one figure 
corresponding with the line or point which is the locus in quo of the range or pencil in 
the other figure. And generally, to any curve of any order and class in the first figure, 
and to its ineunts and tangents, there correspond in the second figure a curve of the same 
order and class, and the ineunts and tangents of such curve. 
195. It is to be remarked, that it is not by any means necessary that the word or the 
words plane, point, and line, or consequently the words order and class, should have the 
same significations as regards the two figures respectively. The theory of homography, 
as above explained, in fact comprises what is commonly termed the theory of homo- 
graphy and also the theory of reciprocity. 
196. Let the word plane have the ordinary signification as regards the two figures 
MDCCCLIX. M 
