564 
A SIXTH MEMOIR UPON QUANTICS. 
[158 
For equations of a higher order the analytical question is considered, and as regards 
the quartic and the quintic respectively completely solved, in my “ Memoir on the 
Conditions for the Existence of given Systems of Equalities between the Roots of an 
Equation ” ( 1 2 ). 
152. Any covariant of the equation 
(*$#> 2/) m = 0, 
equated to zero, gives rise to a point-system connected in a definite manner with 
the original point-system. And as regards the invariants, the evanescence of any 
invariant implies a certain relation between the points of the system; the identical 
evanescence of any covariant implies relations between the points of the system, such 
that the derived point-system obtained by equating the covariant to zero is absolutely 
indeterminate. The like remarks apply to the covariants or invariants of two or more 
equations, and the point-systems represented thereby. 
153. In particular, for the two point-pairs represented by the quadric equations 
(a, b, c\x, yf = 0, 
(a', b', c'$>, y) 2 = 0, 
if the lineo-linear invariant vanishes, that is, if 
ac — 2 bb' + ca' = 0, 
we have the harmonic relation,—the two point-pairs are said to be harmonically 
related to each other, or the two points of the one pair are said to be harmonics 
with respect to the two points of the other pair. The analytical theory is fully 
developed in the “ Fifth Memoir upon Quantics ” ( 2 ). 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 re 
maining 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 
two given 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) sibiconjugate points of the involution. A system of three or more pairs, 
such that the third and every subsequent pair are each of them harmonically related 
to the sibiconjugate points of the first and second pairs, is said to be a system in 
involution. In particular, for three pairs we have what is termed an involution of 
six points ; and it is clear that when two pairs and a point of the third pair are 
given, the remaining point of the third pair can be determined. And in like manner 
1 Philosophical Transactions, vol. cxlvii. (1857), pp. 727—731, [150]. 
2 Philosophical Transactions, vol. cxlviii. (1858), pp. 429—462, [156].