64 
I^IE. A. CAYLEY’S SIXTH JVIEMOIE LPOX QrAXTICS. 
point-system obtained by equating the covariant to zero is absolutely mdetenniuate. 
The like remarks apply to the covariants or invariants of t^^o or more equations, and the 
point-systems represented thereby. 
153. In particular, for the two point-pairs represented by the quadnc equations 
{a,h,cJx,yY^^, 
(a', b', djx, y)"=0, 
if the lineo-linear invariant vanishes, that is, if 
we have the harmonic relation,— the two point-pairs are said to be harmonicaUy related 
to each other, or the two points of the one pah are said to be harmomcs with respect to 
the two points of the other pair. The analytical theoi7 is fully developed m the '■ Fi 
Memoir upon Qualities*.” The chief results, stated under a geometrical form, are as 
follows: — . . 
1°. If either of the pairs and one point of the other pair are given, the remaming 
point of such other pair can be found. 
2°. A point-pair can be found harmonically related to any two given point-pairs. 
154. The last of the two theorems gives rise to the theory of involution. ^ The t^o 
o-iven point-pairs, viewed in relation to the harmonic pair, are said to be an involution 
of four points ; and the points of the harmonic pair are said to be the (double or) sibi- 
coniugate points of the involution. A system of three or more paus, such that the 
third and every subsequent pair are each of them harmonically reiated to the sibi- 
conjugate points of the first and second pairs, is said to be a system in mvoliition. ^ In 
particular, for three pairs we have what is termed an involuUon of six points ; and it is 
clear that when two pairs and a point of the third pair are given, the remaming point^ ol 
the third pair can be determined. And in like manner for a greater number of pairs, 
when two pairs and a point of each of the other pairs are given, the remainmg point ot 
each of the other pairs can be determined. Two points of the same pan- ai-e said to be 
coniugate to each other; or if we consider two pafis as given, then the points of the 
third or any subsequent pair are said to be conjugate to each other in respect to the 
mven pah's. This explains the expression sibiconjugate points ; in fact, the two pairs 
Ling 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 pan- of 
coincident points, form a system in involution, or involution of five points. 
155. The three point-pairs, U = 0, U'=0, U"=0, will be in involution '^^en^ke 
quadrics U, U', U" are connected by the linear relation or syzygy xU-h>^'U'+V'U' _0. 
This property, or the relation 
a , 
b, 
a', 
b', 
a". 
b", 
* Philosophical Transactions, vol. cxlviii. (1858), pp. 4'29-462. 
