644 
PROFESSOR STLYESTER ON THE REAL 
planar regions on opposite sides of the line AO A ; and again this line into two linear 
regions on either side of the origin O, which last corresponds to the case of three equal 
roots, and constitutes a region or microcosm in itself. 
(68) It may as well be noticed here that the ambiguity of character in the points 
representing the different families of biquadratic forms when t and I) are taken as the 
coordinates (and the same would be true if s and D were employed), which prevails 
when these points lie above the line D = 0, equally obtains along this line itself. For 
the reduced form, when D=0, is ax 4, + ibx^y + 6 cxhf. In that case, calling the deter- 
minant of transformation [x, we have s= 3^ 12 <? 2 , D= — [x 2i c 3 ; and thus, whatever s and 
D may be, the character of the unequal roots is left undecided. 
It may also be noticed that the blending of characters at the origin for the quintic 
form is not precisely of the same nature as that for the points above the line D in 
the biquadratic form ; for at these points it is the cases of 4 and 0 imaginaries which 
become undistinguishable invariantively ; whereas at the origin for quintics the re- 
duced form becomes ax s -j- 5 hx 4 y -j- 1 O^y 3 , and the characters left undistinguished are 
those of 4 and of 2 imaginary roots — unless, indeed, we consider equal real roots as 
belonging indifferently to the class of real and imaginary ; on which supposition all the 
three genders (so to say), masculine, feminine, and neuter, become blended together at 
that point. But if we consider equal real roots as exclusively of the real class, then the 
origin for quartics ceases to be epicene ; for when there are three equal roots all of them 
must be real. Thus the origin in quintics is the only epicene point, and in quartics 
the only non-epicene point — understanding by epicene the blending of the masculine 
(4 imaginary roots) and feminine [no imaginary roots ) characters. 
(69) We may draw some further important inferences from an inspection of the 
“ dial figure,” or the section of facultative space which follows it. 
Within the prism JO A' ( 57 ) it will be observed D is always positive ( 58 ). Hence, when 
J is negative and A' is negative, all the roots must be real, and the necessity for using 
the criterion D is done away with. 
Again, when J and L are both negative, D is always negative, so that just two of the roots 
must be imaginary ; and in this case also it becomes unnecessary to apply the criterion D. 
Again, since there is no facultative prism corresponding to IIOJ, the combination 
of L and D, both negative, can never occur unless n is negative. 
When L is negative, but J not negative, there may be two or four imaginary roots, 
according to the sign of D ; but all the roots cannot be real. 
(70) M. Hermite’s rule is as follows. For remarks on the relation between his A, 
J 2 , J 3 and the J, K, L of this paper, see foot-note ( 36 ). D is still the discriminant. 
If D is negative (of course) two roots are imaginary. 
If D is positive. 
( 57 ) By which I mean within the facultative prism of which JOA' is the section made by the plane of D. 
( 58 ) phg vertical seetion of facultative space in this supposition (see figure) is the area ACx', which lies wholly 
above the plane of D. 
