566
A SIXTH MEMOIR UPON QUANTICS.
[158
157. A lineo-linear equation
(x - ay) (x - ay) =0
denotes then the two points (x, y = a, 1) and (x, y = a, 1) existing irrespectively of
each other in distinct spaces, and only by the equation itself brought into an ideal
connexion ; and any invariantive relation between the coefficients of any such bipartite
function denotes geometrically a relation between a point-system in the space which
is the locus in quo of the coordinates (x, y), and a point-system in the space which
is the locus in quo of the coordinates (x, y) ; for instance, the equation
1, a, a, aa
1, 5, /3, 5/3
1, c, y, cy
1, d, 8, d8
= 0
is the relation of homography between the four points (a, 1), (b, 1), (c, 1), (d, 1)
in the first line, and the four points (a, 1), (/3, 1), (7, 1), (8, 1) in the second line.
The analytical theory is discussed in the Fifth Memoir; and, in particular, it is
there shown, that writing
A = (d - a) (5 - c), gt = (8 - a) (/3 - 7),
B = (d-b) (c-a), 23 = (8-£)(7-a),
C = (d — c) (a — b), (& — (8 — 7) (a — /3),
then the condition may be expressed under any one of the forms
A : B : G = & : 23 :
equations which denote the equality of the anharmonic ratios of the two point-systems.
158. The number of points in each system may be four, or any greater number;
the homographic relation is then conveniently expressed under the form
1,
1 ,
1 ,
1 ,
1, ...
a ,
5 ,
0 ,
d ,
e ,
a ,
/3,
7 »
5 ,
e ,
au,
5/3,
cy,
d8,
ee,
The relation is such that given three points of the one system and the corresponding-
three points of the other system, then to any fourth point whatever of the first
system there can be found a corresponding fourth point of the second system. It is
to be observed, however, that two systems of four points homographically related to
each other, always correspond together in four different ways, viz. the two systems
being (a, b, c, d) and (a, ¡3, 7, 8); then if the four points of the first system are
{a, b, c, d), the corresponding four points of the second system may be taken in the
four several orders, (a, /3, 7, 8), (/3, a, 8, 7), (7, 8, a, /3), (8, 7, /3, a).
