ME. A. CAYLEY’S SIXTH MEMOIE UPON QUANTICS. 
65 
to which it gives rise, might have been very properly adopted as the definition of the 
relation of involution, but I have on the whole preferred to deduce the theory of involu- 
tion from the harmonic relation. The notion, however, of the linear relation or syzygy 
of three or more point-systems gives rise to a much more general theory of involution, 
but this is a subject that I do not now enter upon; it may, however, be noticed, that if 
U=0, U'=0 be any two point-systems of the same order, then we may find a point- 
system U"=0 of the same order, in involution with the given point-systems (that is, 
satisfying the condition ^-nd such that the point-system XT 0 
comprises a pair of coincident points ; this is obviously an extension of the notion of the 
sibiconjugate points of an ordinary involution. 
156. It was remarked in the Fifth Memoir, that the theories of the anharmonic ratio 
and of homography belong analytically to the subject of bipartite (lineo-linear) binary 
quantics ; this may be further illustrated geometrically as follows : we may imagine two 
distinct spaces of one dimension or lines, one of them the locus in qiio of the coordinates 
(^, y\ and the other the locus in quo of the coordinates (x, y), which are absolutely inde- 
pendent of, and are not in anywise related to, the coordinates of the first-mentioned 
system. There is no difficulty in the conception of this ; for we may in a plane or in space of 
three dimensions imagine any two lines, and study the relations of analogy between the 
points of the one line inter se, and the points of the other line inter se, without in any- 
wise adverting to the space of two or three dimensions which happens to be the common 
locus in quo of the two lines. It is proper to remark, that in speaking of the spaces of 
one dimension, which are the loci in quibus of the coordinates y) and (x, y) respect- 
ively, as being each of them a hne, we imply a restriction which is altogether unneces- 
sary ; the words line and point may, in regard to the two figures respectively, be used in 
different significations ; for instance, one of the spaces may be a line and the points in 
it paints ; while the other of the spaces may be a point and the points in it lines, or it 
may be a line and the points in it planes. 
157. A lineo-linear equation, 
(^— a^)(x— ay) = 0. 
denotes then the two points {x,y—a, I) and (x, y = a, I) 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 functions 
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 
I, a, 
I, h, 
I, c, 
1, d, 
a, 
act 
i3, 
b(i 
cy 
dl 
