ME. A. CAYLEY’S SIXTH MEMOIE UPON QIJANTICS. 
63 
150. A linear equation, 
is ob^dously equivalent to an equation of the before-mentioned form x, h, and 
represents therefore a point. An equation such as 
=0 
breaks up into m linear equations, and represents therefore a system of m pointe, or 
point-system of the order m. The component points of the system, or the linear 
factors, or the values thereby given for the coordinates, are termed roots. When 
m=l we have of course a single point, when 2 we have a quadric or point-pair, 
when m=3 a cubic or point-triplet, and so on. The point-system is the only figure or 
locus occurring in the geometry of one dimension. The quantic VY ■> when it is 
convenient to do so, may be represented by a single letter U, and we then have U = 0 
for the equation of the point-system. 
151. The equation 
{^Xx, 3/r=o 
may have two or more of its roots equal to each other, or generally there may exist any 
systems of equalities between the roots of the equation, or what is the same thing, the 
system may comprise two or more coincident points, or any systems of coincident points. 
In particular, when the discriminant vanishes the equation will have a pair of equal 
roots, or the system will comprise a pair of coincident points; in the case of the quadiic 
(a, h, c\x^ the condition is 5^=0, or as it may be written, a^h-=-h^c\ in 
the case of the cubic 
(a, 5, c, djx, 
the condition is 
a-d^ _ Ulcd + 4ac^ + - 3d^c^ = 0. 
The preceding is the only special case for a quadric : for a cubic we have besides the 
special case where the three roots are equal, or the cubic reduces itself to thiee coin- 
cident points ; the conditions for this are 
ac—b^=0, ad—hc—^, 
equivalent to the two conditions 
a : h=-h : c=-c : d. 
For eqiuations of a higher order the analytical question is considered, and as regaids 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^. 
152. Any covariant of the equation 
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 invaiiant 
implies a certain relation between the points of the system ; the identical evanescence of 
any co variant implies relations between the points of the system, such that the deiived 
* Philosophical Transactions, vol. cxlvii. (1857), pp. 727- 31. 
