156 
AN EIGHTH MEMOIR ON QUANTICS. 
[405 
contains each root in the power 18, and is consequently a rational and integral 
function of the coefficients of the degree 18, viz. save as to a numerical factor it is 
equal to the invariant I. And considering the equation (a, . .\x, y) 5 = 0 as representing 
a range of points, the signification of the equation 1 = 0 is that, the pairs (/3, 7) and 
(8, e) being properly selected, the fifth point a is a focus or si biconjugate point of 
the involution formed by the pairs (/3, 7) and (8, e). 
Article Nos. 257 to 267.—Theory of the determination of the Character of an 
Equation; Auxiliars; Facultative and Non-facultative space. 
257. The equation (a, b, c ...][x, y) n = 0 is a real equation if the ratios a : b : c,.. 
of the coefficients are all real. In considering a given real equation, there is no loss 
of generality in considering the coefficients (a, b, c..) as being themselves real, or in 
taking the coefficient a to be = 1 ; and it is also for the most part convenient to 
write y = 1, and thus to consider the equation under the form (1, b, c...\x, l) n = 0. 
It will therefore (unless the contrary is expressed) be throughout assumed that the 
coefficients (including the coefficient a when it is not put =1) are all of them real; 
and, in speaking of any functions of the coefficients, it is assumed that these are 
rational and integral real functions, and that any values attributed to these functions 
are also real. 
258. The equation (1, b, c..Afx, 1 )‘ l = 0, with a real roots and 2/3 imaginary roots, 
is said to have the character ar+2fii; thus a quintic equation will have the character 
or, ‘Sr + 2i, or r + 4>i, according as its roots are all real, or as it has a single pair, or 
two pairs, of imaginary roots. 
259. Consider any m functions (A, B, ... K) of the coefficients, (m = or < n). For 
given values of (A, B, ... K), non constat that there is any corresponding equation (that 
is, the corresponding values of the coefficients (6, c, ...) may be of necessity imaginary), 
but attending only to those values of (A, B, ... K) which have a corresponding equation 
or corresponding equations, let it be assumed that the equations which correspond to 
a given set of values of (A, B,...K) have a determinate character (one and the same 
for all such equations): this assumption is of course a condition imposed on the form 
of the functions (A, B,...K)\ and any functions satisfying the condition are said to 
be “auxiliars.” It may be remarked that the n coefficients (b, c, ...) are themselves 
auxiliars; in fact for given values of the coefficients there is only a single equation, 
which equation has of course a determinate character. To fix the ideas we may con 
sider the auxiliars (A, B,...K) as the coordinates of a point in m-dimensional space, or 
say in m-space. 
260. Any given point in the m-space is either “ facultative,” that is, we have 
corresponding thereto an equation or equations (and if more than one equation then by 
what precedes these equations have all of them the same character), or else it is 
“ non-facultative,” that is, the point has no corresponding equation. 
261. The entire system of facultative points forms a region or regions, and the 
entire system of non-facultative points a region or regions; and the m-space is thus 
divided into facultative and non-facultative regions. The surface which divides the