AND IMAGINARY ROOTS OR EQUATIONS. 593 
within the curve A will correspond to real values of s, q. In order that these quantities 
may be real, we must have 
••+«•> 2 («,)* 
i. e. <r-\-q 2 +q 3 >2qf I , where q=p + 1, 
or 
(d-\-2(q 2 -\-q 3 )G+q*— 4^ s +g 6 >0. 
Writing this inequality under the form R>0, we see that the curve R=0 will repre- 
sent a second sextic curve intersecting the former. A may be called the curve of 
the discriminant or discriminatrix, and will be a close curve, and R the curve of equal 
parameters or equatrix, and will consist of a single infinite branch. All points on the 
latter correspond to equal values of s, q, those on one side of it to real values of s, q, 
and those on the other side of it to conjugate values of the form a +?//,, X—ip respectively. 
Thus the area confined within the curve A will be divided into two portions by the 
equatrix, and it is impossible to shut one’s eyes to the inquiry as to the meaning of the 
variable point lying in that portion which gives conjugate values to s, q. It becomes 
clear by analogy that some kind of distinction must be capable of being drawn between 
the nature of the roots of the equation (1, s, s 2 , q\ q, l^x, y) 5 = 0 when s, q are conjugate, - 
in some sense similar or parallel to that which we know to exist between them when s, q 
are real ; and obviously this inference cannot be confined to equations of the particular 
form and degree of that above written ; in a word, equations whose coefficients are not real 
but conjugate, must have roots of two kinds, one analogous to the real, the other to the 
imaginary roots of equations with real coefficients. This inference will be justified 
in the sequel ; but in the meanwhile it will be desirable to complete the investigation 
of the special equation under consideration, by a discussion of the forms and relations 
of the two curves A and R. These curves we know a priori^ from what has been already 
demonstrated, can only meet in the three points corresponding to 
e=q—l, i=q=%, ; 
and since p=sq — 1, the abscissae of these three points will be 0, — j, — f. 
Moreover the 3rd point will be distinguished from the other two by the circumstance 
that A does not change its sign as p passes through the value — f. Consequently 
the two curves must touch each other at this point. 
Since when A = 0 y? lies between 0 and — the curve A is confined to the negative 
side of the axis of a. It is also confined to the negative side of the axis of p. 
For between the limits p — 0, p = — , 
648p 2 +672p 3 , i. e. 24(27j» 2 +28j? 3 ) is obviously positive, 
and 
96j? 6 +176p 5 -|-81^> 4 = ~-{(24^)+22) 2 -{-2} is always positive. 
Hence the two values of a are both negative throughout the extent of the curve A. 
Thus e 5 +*7 5 — sV— sV being negative, s 3 — q 2 and q 3 — s 2 have the same signs when g, q 
