AND IMAGINARY ROOTS OF EQUATIONS. 
643 
the plane of D no change can possibly take place except in crossing the line which 
corresponds to a family of two pairs of equal roots. 
(67) It has already been pointed out, in a foot-note, that we cannot pass facultatively 
from OA to either side of this curve line. Hence the separation of the plane of D into 
subregions can only take place along the line OA, and it remains but to ascertain the 
character of the points on either side of this line, which we know, therefore, a priori, 
must possess opposite characters, since otherwise we should be admitting the absurd 
proposition of its being impossible to construct an equation of the fifth degree having 
two equal roots without the remaining three being always of one character , either all 
real or all not real. Let us, then, ascertain the character of the points in OJ for which 
D=0, L=0, and J is positive ( 56 ). 
Since L=0, the reduced form is u 5 -^beuv* -\-v 5 . 
This equation, by Descartes’s rule, must contain imaginary roots. Hence in the sector 
AOJ the roots are all real, and in the remainder of the facultative portion of the plane 
(from which it may be noticed the sector AOJ is excluded) two of the roots are imagi- 
nary. 
Along OA itself there are, as already observed, two pairs of real equal roots, and 
along OA two pairs of imaginary equal roots. Thus, finally, we have the complete rule. 
If D is negative, 2 roots imaginary. 
If D is positive. 
When J, A are both negative, 0 roots imaginary. 
„ J, A are not both negative, 4 roots imaginary. 
If D is zero. 
When J, A are both negative, 0 roots imaginary! . „ , 
t a , , ,, , . 0 , . . 11 pair of equal roots. 
„ J , A are not both negative, 2 roots imaginary J 
„ J is negative, A zero, 0 roots imaginary! _ . . 
T . ... . , , . . 12 pairs of equal roots. 
„ J is positive, A zero, 4 roots imaginary J 
„ J is zero, A zero, 3 equal roots( 56bi3 ). 
Thus we see that our space referred to an arbitrary origin, and with the invariants 
J, D, A for the coordinates, has been first divided into facultative and non-facultative 
space. The former has then been resolved prismatically into two regions above and one 
below the plane of D. The plane of D itself, or the facultative part of it, into two 
(56) yy e could not take J negative, for the facultative points of D in J are two positive quantities. See dial 
figure. 
(56 bis) D = o, A=0, there are two pairs of equal roots. If J is negative these pairs are both real. If 
J is positive they are both imaginary. When J is zero there are no longer two pairs, but a single triad of equal 
roots. This perfectly explains what at first sight has the air of a paradox, viz. that the discrimination between 
the two kinds of double equality of an apparently equal order of generality that may subsist between the roots 
of an equation, depends on the fulfilment or failure of an algebraical equality. The fact is, as shown above, 
that there are not, as commonly supposed, two, but three kinds of double equality, according as there are two 
pairs of real, two pairs of imaginary, or one triad of equal roots ; and the last is a sort of transition case between 
the other two. 
4 R 
MDCCCLXIY. 
