A SIXTH MEMOIH UPON QUANTICS. 
565 
[158 
158] A SIXTH MEMOIH UPOH QUANTICS. 565 
regards 
on the 
! of an 
for a greater number of pairs, when two pairs and a point of each of the other 
pairs are given, the remaining point of each of the other pairs can be determined. 
Two points of the same pair are said to be conjugate to each other; or if we 
r with 
if any 
entical 
1, such 
olutely 
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consider two pairs as given, then the points of the third or any subsequent pair are 
said to be conjugate to each other in respect to the given pairs. This explains the 
expression sibiconjugate points; in fact, the two pairs being given, either sibiconjugate 
point is, as the name imports, conjugate to itself. In other words, any two pairs 
and one of the sibiconjugate points considered as a pair of coincident points, form a 
system in involution, or involution of five points. - ; 
155. The three point-pairs, U = 0, U' — 0, TJ" = 0, will be in involution when the 
quadrics TJ, U', TJ" are connected by the linear relation or syzygy \ TJ + V TJ' + TJ" = 0. 
This property, or the relation 
a , b , c = 0 
a!, b', d 
a", b", c" 
ns 
nically 
nonics 
fully 
ider a 
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 theor} 7 
of involution 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 TJ" = 0 of the same order, in involution with the given 
point-systems (that is, satisfying the condition \TJ + \'TJ' + X"TJ" = 0), and such that 
the point-system TJ" = 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 
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ratio and of homography belong analytically to the subject of bipartite (lineo-linear) 
binary qualities; this may be further illustrated geometrically as follows: \ve may 
imagine two distinct spaces of one dimension, or lines, one of them the locus in quo 
vs. 
of the coordinates (x, y), and the other the locus in quo of the coordinates (x, y), 
which are absolutely independent of, and are. not in anywise related to, the co 
The 
be an 
e the 
ordinates of the first-mentioned system. There is no r 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 anywise adverting to the space of 
pairs, 
elated 
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 
m in 
on of 
are the loci in quibus of the coordinates (x, y) and (x, y) respectively, as being each 
of them a line, we imply a restriction which is altogether unnecessary; the words 
r are 
line and point may, in regard to the two figures respectively, be used in different 
anner 
significations; for instance, one of the spaces may be a line and the points in it 
points; 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.