158] 
A SIXTH MEMOIR UPON QUANTICS. 
567 
159. What precedes is not to be understood as precluding the existence of a 
relation between the spaces which are the loci in quibus of the coordinates (x, y) 
and (x, y) respectively: not only may these be spaces of the same kind, but they 
may be one and the same space or line; and the points of the two systems may 
then be points of the same kind; and further, the coordinates (x, y) and (x, y) 
may belong to the same system of coordinates, that is, the equations {x, y = a, 1) 
and (x, y = a, 1) may denote one and the same point. 
160. If the two point-systems are systems of the same kind, and are in one 
and the same line, then there are in general two points of the first system which 
coincide each of them with the corresponding point of the second system; such two 
points may be said to be the sibiconjugate points of the homography. In particular, 
the two sibiconjugate points of the homography may coincide together. 
161. A system in involution affords an example of two homographic systems in 
the same line; in fact, taking arbitrarily a point out of each pair, the points so 
obtained form a system which is homographic with the system formed with the other 
points of the several pairs; and in this case the sibiconjugate points of the involution 
are also the sibiconjugate points of the homography. Thus if A and A', B and B', 
C and O', D and D' are pairs of the system in involution, then (A, B, C, D) 
and {A', B', C', D') will be homographic point-systems; and, as a particular case, 
{A, B, G, C') and (AB', O', C) will be homographic point-systems. It is proper 
to notice that if F is a sibiconjugate point of the involution, then {A, B, F, F) 
and (A', B', F, F) are not (what at first sight they appear to be) homographic 
point-systems. 
162. Imagine an involution of points; take on the line which is the locus in 
quo of the point-system a point 0, and consider the point-system formed by the 
harmonics of 0 in respect to the several pairs of the involution; and in like manner 
take on the line any other point O', and consider the point-system formed by the 
harmonics of 0' in respect to the several pairs of the involution; these two point- 
systems are homographically related to each other.—See Fifth Memoir, No. 111. 
163. Two involutions may be homographically related to each other; in fact, 
take on the line which is the locus in quo of the first involution a point 0, and 
consider the point-system formed by the harmonics of 0 in relation to the several 
pairs of the involution; take in like manner on the line which is the locus in quo 
of the second involution a point Q, and consider the point-system formed by the 
harmonics of Q with respect to the several pairs of the involution; then if the two 
point-systems are homographically related, the two involutions are said to be them 
selves homographically related: the last preceding article shows that the nature of 
the relation does not in anywise depend on the choice of the points 0 and Q. 
And it is not necessary that, as regards the two involutions respectively, the words 
line and point should have the same significations.—See Fifth Memoir, No. 111. 
164. Four or more tetrads of points in a line may be homographically related 
to the same number of tetrads in another line. This is the case when the an-