AND IMAGINARY ROOTS OR EQUATIONS. 
641 
Let g=+l+<r, where a is an infinitesimal. Hence 
(_10(+/-l) 5 +10(+/+l) 5 y-8s=0, 
or 
20(1— 10+5)<r 2 — 8e=0, 
or 
ff2= io e= +Hloo B (r)- 
Hence calling <r„ <r 2 the two values of <r, the four roots that at OA were 1, 1, —1, — 1 
become 1+<7 1? 1 -f-o- 2 , — l-J-<7], — l-fff 2 , when jj becomes varied by and conse- 
J 3 . _ J3 
quently become all real if jj is increased, and all imaginary if jj is decreased, i. e. be- 
come real or imaginary according as the line OA sways towards or away from OJ, con- 
formably with what has been shown on other grounds. 
It will be noticed that in the line OA produced in the opposite direction, i. e. along 
the line OA, L being positive, the reduced form is 
4(w 5 + v 5 ) + ( u + v) 5 = 0, 
and the roots of - become -= — 1, -= +/, -= +/ : so that, according to the canon laid 
down at the commencement of this discussion (see foot-note ( 46 )), no change in the cha- 
racter of the roots can possibly take place along OA, and accordingly we have seen that 
this curved line does not correspond to any demarcation of regions. 
(65) It is easy to express the conditions to be satisfied by the coordinates of a point 
according as it lies in one or another of the three regions which have now been mapped 
out, and it is clear that' we have the following rule : 
When D is negative the equation has two imaginary roots. 
When D is positive the equation has no imaginary roots, provided the two criteria J and 
2 U L — J 3 are both negative( 53 ); but if either of these is zero or positive, there are two 
pairs of imaginary roots ( 54 ). 
The duodecimal criterion-invariant, 2 n L — J 3 , and the invariants of the like order, 
27'2 l6 L— J 3 , — 27L— J 3 , I shall henceforth call A, A', n respectively. It has been just 
above shown that the three invariants J, D, A of the 4th, 8th, and 12th orders re- 
spectively are sufficient for ascertaining the character of the roots of the quintic to 
which they appertain. 
J 3 J3 
( 53 ) Observe that this implies L also being negative ; so that 2 11 — -- is positive and — «<2 U . 
Ju -L 
( 54 ) (a) Observe that in general when 2 n L— J is zero there are no facnltative points above the plane of D, but 
when J and 2 n L— J, and consequently L and J are both simultaneously zero, a facnltative right line springs 
into existence, viz. the axis of D extending both above and below the plane of D. The reduced form of equa- 
tion (as previously demonstrated) corresponding to this singular line is u i +uv i =0. 
( b ) It may further be noticed that on each side of the line OA the limits of D are between positive infinity 
and a positive quantity, and between negative infinity and a negative quantity ; so that as we pass from OA to 
either side of it no facultative point can be found lying in the plane of D, showing that we cannot pass by a 
real infinitesimal variation of coefficients from an equation with two pairs of equal imaginary roots to an equa- 
tion with a single pair of equal roots, as is apparent also on purely analytical grounds. 
