AM) IMAGINARY ROOTS OF EQUATIONS. 
595 
There is a third curve not undeserving of notice, of only the 3rd degree, which 
embodies the joint effect of the two middle criteria (the two extremes being supposed 
to be each zero) in the two cases where Newton’s rule will prove all the roots of the 
equation under consideration to be impossible. These criteria are Cj=s 4 — £?? 3 , c 2 —r? — * 7 s 3 . 
But c 1 ,‘+ c /= 2 (2 2 s -S)= 2 (2 2 ! - 2 >- 2 a -*)= 2 ( 2 a - 2 a -,x), 
which for all values of q on the positive side of the line p— — \ (i. e. ^= 0 ) will have the 
same sign as q 3 — q 2 — <r, which we may call K( 20 ); and K positive will evidently imply 
that c„ c 2 are one or both of them positive. The whole plane will be divided by the 
curve K into an upper region (commencing at a = 00 ), for which K is negative, and a 
lower region, in which K is positive. For any point of the curve K, a=q 3 — q 2 , which 
within the limits of q with which we are concerned, viz. those within which A lies, 
is negative ; for any point of the curve R, the smaller absolute value of <7 is 
-<?-?+ 2 2 i= 2 a - 2 a +2( 2 t- 2 a ), 
which <q 3 — q 2 within the limits in question. So that, remembering that each of these 
values of a is negative, we see that the portion of the area A corresponding to real values 
of s, n will be completely above the curve K, i. e. in the negative region of K, and that 
accordingly A for real values of s, >7 can never vanish when K is positive, as should be 
the case. This remark does not, however, apply to the conjugate region of A ; for the 
curve K will pass through ( 21 ) the lower or conjugate portion of the area A. 
(12) I may now say a few words on the signification of that portion of A in which s 
and 7j are conjugate imaginary quantities. 
of a quadratic and the extraction of 5th roots. To find the maxima and minima values of s and y themselves 
exactly would lead to the solution of an equation of a degree quite unmanageable. 
But we may first find the greatest maximum and least minimum values of S, i. e. £°+ij s , by making 
Str= (2q-\-3q*)8q in 5A=0, which leads to an equation (I forget whether) of the 3rd or 5th degree (it is one of 
the two) : calling this maximum and minimum m, fx respectively, and naming ^ (which 'of course must exceed 
unity) the greatest quotient of - or we shall have 
V s 
i+p"^ E! ’=-\/ i t ?*- 
These limits will he tolerably near to the absolute maximum and minimum values of s or y. It may be noticed 
that we know, from what has gone before, that j> can never exceed 5 ; and consequently cannot exceed 4, 
since q is always >1. 
( 20 ) I call K the Indicatrix, as exhibiting the joint effect of the indicia or criteria of the Rule. 
( 21 ) This may easily he verified ; for at the point p = — J it will be found that the ordinate in K and the lower 
ordinate in A are equal, and at the point p= — -fa the lower ordinate in A is — gwow, and in K is —z hon > 
which shows that the curve K entering the area A when at the lower half of the curve, at a point where p= — f, 
must pass through its upper contour in order to cut the line p= — T 9 7 as it does above the point where A 
is touched by that line. 
The curve K has its negative maximum at the point <?=■§•, i. e. p— — 1. It passes through the origin, and 
begins with sweeping under the curve A, which it enters exactly under the point where R quits A, and passes 
MDCCCLXIV. , 4 L 
