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PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
258. The equation (1, b, c . . .Jjv, 1)”=0, with a real roots and 2/3 imaginary roots, is 
said to have the character ar-\-2j3i ; thus a quintic equation will have the character 5 r, 
3r+2i, or r+4«, 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 (#, c , . . .) may be of necessity ima- 
ginary), 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 
consider 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 corre- 
sponding 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-faculta- 
tive,” 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 facultative 
and non-facultative regions may be spoken of simply as the bounding surface, whether 
the same be analytically a single surface, or consist of portions of more than one 
surface. 
262. Consider the discriminant D, and to fix the ideas let the sign be determined in 
such wise that D is -f- or — according as the number of imaginary roots is = 0 (mod. 4), 
or is = 2 (mod. 4); then expressing the equation D = 0 in terms of the auxiliars 
(A, B, . . . K), we have a surface, say the discriminatrix, dividing the m-space into regions 
for which D is -}-, and for which D is — , or, say, into positive and negative regions. 
263. A given facultative or non-facultative region may be wholly positive or wholly 
negative, or it may be intersected by the discriminatrix and thus divided into positive 
and negative regions. Hence taking account of the division by the discriminatrix, but 
attending only to the facultative regions, we have positive facultative regions and nega- 
tive facultative regions. Now using the simple term region to denote indifferently a 
positive facultative region or a negative facultative region, it appears from the very 
notion of a region as above explained that we may pass from any point in a given region 
to any other point in the same region without traversing either the bounding surface or 
the discriminatrix ; and it follows that the equations which correspond to the several 
