634 
PROFESSOR SYLVESTER ON THE REAL 
(54) This is precisely what happens in biquadratic equations. In such we know the 
fundamental invariants t, s, or, if we please, t, A (where A=s 3 +27£ 2 ), are perfectly inde- 
pendent and subject to no equation of condition; so that if we consider t , A as the 
coordinates of points in a plane, the whole of the plane will be made up of facultative 
points. When A is negative, i. e. for representative points lying on one side of the line 
A, it is true we know that there is just one pair of imaginary roots constituting what 
may be termed the neuter case ; but when the representative points lie on the other side 
of this plane, they cannot be said to be either masculine or feminine, but will every one 
of them possess that epicene character which is peculiar to the origin alone in the case 
of quintic forms. A single example will make this clear. 
Take the two reduced forms 
u 4 + 6 ( 1 + s) w V + v 4 , 
a 4 +6(l-g> 2 0 2 -H 4 , 
where u, v are real linear functions of x, y, and a, 6 conjugate imaginary ones of the 
same ; and suppose s, the quadrinvariant in respect to x,y, to be the same for both forms. 
For greater convenience of computation consider g to be infinitesimal. 
Then in the one case the t is of the same sign as 
(l +f )(l — (l +£ )=),i.e. -26, 
and in the other the t is of the contrary sign to 
(1 — g )(l — (1 — s ) 2 )’ i- e - 26, 
so that t is of the same sign (viz. negative) in each case. 
Again, in the two cases respectively 
s 3 ~ l+3(l+e) 2 “ 4£ * 
Hence t as well as s, and consequently t and A are alike for both forms. 
But in the one first written the roots are of the same nature as those of u 4 -\ -6wV-{-?; 4 ,' 
i. e. are all impossible, and in the other of the same nature as in 
where u, v are real linear functions of x, y and i=\/ —1, in which case the roots are 
all possible. Thus we see that the very same values of t , A may correspond either to 
the case of four real or four imaginary roots, showing that the point t , A is what we 
have termed epicene. If we choose to take s, t as the coordinates, the same remarks 
would apply, except that A instead of a straight line would become a semicubical para- 
bola. All the points on one side of this curve would have a definite neuter character, 
but those on the opposite side would be neither masculine nor feminine, but epicene. 
(55) With a view to its subsequent distribution into regions, I now proceed to ascertain 
the form of that moiety of space which I have termed facultative. 
