66 
ME. A. CAYLEY’S SIXTH I^IEMOIE TPOX QrAXTICS. 
is the relation of homography between the four- points {a, 1), {h, 1), e, 1), m the 
tot line, and the four points (a, 1), (/3, 1), ('/, 1), (S, 1) m the second toe. ^e analy- 
tical theory is discussed in the Fifth Memoir ; and, in particular, it is there shown, that 
A=(d-a)(6-c), a = (5-“)0-'A 
B={cl~b){c~a), §S = (B--/3Xy"«), 
C=\d—c){a~h), C=0-7)(oi-/3), 
then the condition may be expressed under any one of the fonns 
A : B : C = a : B : C, 
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 , e 
homographic relation is then conveniently expressed under the form 
1,1, 
1 , 
1 , 
1, 
« , b , 
c , 
d , 
6, 1 
a , /3 , 
y , 
2 , 1 
««, 5|3, 
cy. 
d\ 
! 
The relation is such that given three points of the one system and the corresponding 
three points of the other system, then to any forrrth point whatever of the &st sptem 
there can be found a correspondirrg fourth point of the second system. t rs to e 
observed, however, that two systems of four poirrts homographically related to each 
other, always correspond together in four different ways, Az. the two systems bemg 
(a, h, c, d) and (a, y, ; then if the four points of the fir-st system are {a, b, c d) t m 
corresponding four points of the second system may be taken rn the forrr several orders. 
{«, /3, y, §), (f3, a, y), (y. ^3), 0, y, i 
159. What precedes is not to be understood as precluding the existence o a^ie a i > 
between the spaces which are the loci in quibiis of the coordmates and (x, y) respect- 
ively : not only may these be spaces of the same kind, but they may be one and 1 
space or line; and the points of the two systems may then be points of the same kmc ; 
and further, the coordinates (x, y) and (x, y) may belong to the same system of coorc i- 
nates, that is, the equations (^, y=a. 1) and (x, y=«, 1) may denote one and the same 
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 tuo points inai e 
said to be the sibiconjugate points of the homography. In particular, the two sibicon- 
iugate points of the homography may coincide togethei. ^ ^ 
161 A system in involution affords an example of two homographic systems in le 
same line; in fact, taldng arbitmrily a point out of each pair, the points so obtamec 
form a system which is homographic with the system formed with the ot lei pom s o 
