582 
PROCESSOR SYLVESTER ON THE REAL 
cients of any function F of two degrees lower than f whose roots are also all real, be 
2>!, jPn - 15 the quadratic function p^Q\ +P 2 Q 2 + • •• +jV-iQ»-i must have its roots 
real, i. e. its discriminant must be positive : a particular consequence of this is, that by 
causing F to consist successively of the single terms x n ~ 2 , x n ~ 3 y, .... xy n ~ 3 , y n ~ 2 , we see 
that the determinants of Q,, Q 2 , ... Q„_, must each of them be positive; or, in other 
words, if any of the Newtonian criteria of an equation are negative, it must have some 
imaginary roots, which is all that Maclauein, Campbell, and others have succeeded in 
proving. 
(4) The labour of proof of the cases hereinafter considered will be much lightened by 
the following rule of induction, viz., granting Newton’s rule to be true for the degree 
n — 1 , it must be true for all those cases appertaining to the degree n in which the series 
of the signs of the criteria does not commence with — j- and end with -\ : to prove 
this, we have only to remember that f must have at least as many imaginary roots as 
% or and that the criterion-series corresponding to and to ~ will be found by 
dx dy ^ & dx dy J 
cutting off from the series off one term to the right and left respectively ( 8 ). If, now, 
the series for f begins with ++ or or -j — , the number of negative sequences is 
the same as when the left-hand sign is removed ; so that it is only necessary to prove that 
the number of imaginary roots in / 1 is not less than the number of negative sequences in 
but this, by hypothesis, is not greater than the number of pairs of imaginary roots 
in and, a fortiori, not greater than the number of such in f. In like manner, if 
the two last criteria off are not -\ — , it may be shown that the truth of the rule for 
such form of/ 1 is implied in what is supposed to be known to be true for 
We may therefore limit our attention, as we ascend in the scale of proof, to those 
forms of/* in which the criterion-series begins with 1- and ends with -j . Accord- 
ingly, since the rule is a truism for n— 2, it is at once proved, by virtue of the above 
considerations, for w=3( 9 ). 
( s ) E° r ^ (a, b, . . . Tc, Tfx, y) n = n(a, b, ... 7c fx, yf 1 , 
and 
jy(a,b, ... Tc, Vfx, y) n =n[ b, ...Tc, TJx, y) n ~ l . 
( 9 ) The theorem for the case of cubic equations may he also proved directly as follows : 
Writing the equation ax 3 -\-3bx 2 y-\-3cxy 2 +dy 3 =0, the two criteria are L =b 2 — ac, M=c 2 — bd; and the 
discriminant is a 2 e? 2 + 4ae 3 + 4c2& 3 — 36V— 6abcd= A. 
1. Let L and M he of opposite signs, so that one and only one of them is negative. Then 
&.=(ad—bcf — 4 (b 2 —ac)(c 2 —bd)=(ad—bcy— 4LM, 
and is therefore positive. 
2. Let L and M he both negative. The equation may evidently, hy writing x and y for aix, diy, be brought 
under the form „ „ 
x 3 + 3s x 3 y + 3i)xy 3 +y 3 = 0, 
with the conditions e 2 <i j, ij 2 < g ; from which we may deduce that e and y are both positive, and ey<l and >0. 
