AND IMAGINARY ROOTS OR EQUATIONS. 657 
in which J, K, L constitute a non-facultative one. But the curve above traced is obvi- 
for G is the resultant of 
J 2 (Q+4)^ 
ously a homographic derivative of that line 
L I 
J 3 “ (0 + 4)0 3 , 
Hence this latter curve will also separate systems of values of J, D, A corresponding 
to facultative from those corresponding to non-facultative points. Moreover when J is 
negative and D positive, it has been shown (see dial figure) that the values of D (in 
facultative systems) corresponding to finite values of J are limited in magnitude ; hence, 
upon the same suppositions, facultative systems of J, D, A will correspond to the inte- 
rior and contour of the curve we have been considering. 
(89) Accordingly, since D is supposed positive, our sole concern will be with the 
curvilinear triangle CMO and the infinite sector QOX, and we have to show that for all 
points not exterior to those areas A and A-j-^JD have the same sign; that is to say, 
1-f^, or lfi-^ | is positive. 
When y and x have opposite signs (as is the case in the triangle CMO), all negative 
values of | to, and when y and X have the same signs (as is the case in the sector XOQ)> 
all positive values of (m obviously make 1 4 -^ - positive. But furthermore which is 
— 1 for the line OC, is greater than —1 for all points in the triangle just named; and 
again, |, which is \ for OB (the parallel to the asymptote through O), will be less than \ 
for all points in the sector QOX. Thus, then, as regards points either in the triangle or 
in the sector, is always intermediate between — 1 and \ ; so that when ^ lies between 
1 and — 2, - will be always positive, and A and A-f-^JD will bear the same sign O, 
so that A+pJD may be used to replace A as a criterion. Q.E.D. 
(90). It is apparent from the nature of the preceding demonstration that A may be 
replaced by an invariant containing not one merely, but an infinite number of arbitrary 
constants (limited), provided we are indifferent to the degree which the substitute for A 
may assume. To this end we have only to draw any algebraical curve F(#, y )= 0 passing 
through the origin, and with its parameter subject to such conditions of inequality as 
will ensure the mixtilinear triangle and sector COM, XOQ lying on opposite sides of 
the curve. If its degree be n , the number of parameters in F left arbitrary within 
limits will be n ^”~~ 2 , and eF( A, JD), where s means one of the two quantities + 1 or — 1, 
may be used as a criterion in lieu of A. For instance, a common parabola with its axis 
coincident with that of x and passing through O will obviously serve as a screen between 
these figures ; its equation will be y 1 — #=0, and the invariant D 2 — JA, which is of the 
sixteenth degree in the coefficients, will serve together with J and D to fix the nature of 
the roots ; so in general we may obtain invariants of any degree of the form 4 i from twelve 
