592 
PEOFESSOR SYLVESTER ON THE EEAL 
showing, agreeably with what was seen in the Lemma, that the discriminant of 
(1, £, s 2 , s 2 , e, lXar, yf 
vanishes then, and then only, when 
6=1, or «=£, or £=— f, 
but does not change its sign, except as £ passes through the limits 1 and and only 
within those limits can become negative ( 17 ). 
(11) Although the theory of the possibility of the roots of (1, s, £ 2 , ?? 2 , tj, ljjv, y) 5 ^=0 
has now been completely investigated, so far as is necessary for the proof of Newton’s 
theorem applied to equations of the fifth degree, it will be found that the labour will not 
be ill spent of considering more closely the real nature of the criteria which separate 
the case of one pair from that of two pairs of impossible roots in the above equation. 
Newton’s criteria being constructed so as to cover every possible case for equations of 
every degree, will always be found to fit loosely, so to speak, upon each case treated 
'per se ; so that more precise conditions can be assigned in each particular case than those 
which are furnished by his rule. So, ex. gr., it may be remembered that in the equation 
(1, e, e\ e , l\x, y) 4 =Q, Newton’s rule implies only that when e>l, the roots are all 
impossible; but we have found further that unless l>e>j$ (a much closer condition), 
the same thing takes place. 
It is obvious from what has been demonstrated above, that if we treat p and <r, which 
are respectively £?j — 1 and £ 5 +*? 5 — sV— £V, as the abscissa and ordinate of a variable 
point in a plane, the curve A = 0, i. e. (108<r+27 j p 2 -l-28p 3 ) 2 + 72j? 5 + 80^ 6 =0 will be 
the line of demarcation between those values of £, n which correspond to one pair, and 
those which correspond to two pairs of imaginary roots. 
For all values of £, jj corresponding to internal points of the curve A there will be two 
imaginary and three distinct real roots ; for all such as correspond to external points 
there will be four imaginary roots, and for points on the curve two imaginary and two 
equal roots. 
The curve A is a curve of the 6th degree whose form will presently be discussed. 
But there is an important remark to be made in the first instance. Not all the points 
( 1? ) In general the case of equal roots of an equation is the state of transition of two real roots into imaginary, 
or vice versa. But we see by the above instance that this is not necessarily the case always, for A vanishes on 
making e= — y, and two roots become equal without any change in the nature of the roots when s passes 
from being greater to being less than — r. In such case, however, there is a sort of unstable equilibrium in 
the form of the equation, by which I mean that the effect of any general infinitesimal change performed upon 
the coefficients of the equation would be either to cause the real roots in the neighbourhood of e= — i to dis- 
appear by the factor (s + i) 2 becoming superseded by a quadratic function of e with impossible roots, or else a 
region in the neighbourhood of s— — -i- would reappear, for which the equation would acquire two real roots, 
owing to (e + y) 2 becoming superseded by a quadratic function of s with real roots, in which case there would 
be two values in the neighbourhood of e= — for each of which there would be a pair of equal roots in tbe equa- 
tion. Tbe above is probably the first instance distinctly noticed of this singular obliteration of tbe usual effect 
upon real and imaginary roots of a passage through equality, owing to the appearance of a square factor in the 
discriminant. 
