ME. A. CAYLEY’S SIXTH MEMiOIE UPON QUANTICS. 
67 
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 C', 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, C, C') and (A , B , C , 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 guo of 
the point-system a point O, and consider the point-system formed by the harmonics of 
O 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 O' in 
respect to the several pairs of the involution; these two point-systems are homogra- 
phically 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 O, and consider the point- 
system formed by the harmonics of O 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 themselves homographically related : the last 
preceding article shows that the nature of the relation does not in anywise depend on 
the choice of the points O 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 anharmonic ratios 
of the tetrads of the first system are homographically related to the anharmonic ratios 
of the tetrads of the second system. And it is not material which of the three anhar- 
monic ratios of a tetrad of either system is selected, provided that the same selection is 
made for each of the other tetrads of the same system. The order of the points of a 
tetrad must be attended to, but there are in all four admissible permutations of the 
points of a tetrad, viz. if A, B, C, D are the points of a tetrad, then (A, B, C, D), 
(B, A, D, C), (C, D, A, B), (D, C, B, A) may be considered as one and the same tetrad. 
Any three tetrads whatever in the second system may correspond to any three tetrads 
of the first system ; and then given a fourth tetrad of the first system, and three out of 
the four points of the corresponding tetrad of the second system, the remaining point of 
the tetrad may be determined. The words line and point need not, as regards the two 
systems of tetrads respectively, be understood in the same significations. — See Fifth 
Memoir, No. 112. 
165. The foregoing theory of the harmonic relation shows that if we have a point-pair 
{a, h, cjx, 2 /)"= 0 , 
