New Criteria for the Imaginary Roots of Equations. 187 
of the equation, by the method of Budan, leaves only one in- 
terval in doubt: the labour of examining this interval is thus 
spared. 
The criteria exhibited above are only two of a group of 
n—1 criteria, arising from applying the second of those two 
to the several limiting equations derived from the primitive 
equation. By applying the formula [1.] to these derived 
equations, we shall merely obtain a repetition of the same in- 
equality: for, as may be easily shown from the nature of dif- 
ferentiation, if a,b,c be the first three coefficients, either in 
the primitive or in any derived equation, m being the degree 
of that equation, the ratio 
mac 
(m—1) 6? 
will be constant. But if the formula [2.] be applied to the 
successive derived equations, we shall be led to the following 
distinct conditions; where, for uniformity, A, is put for A, 
and A, for N: 
2nA,A,> (n—1)A,? 
3(n—1) A, As>2 (n—2) A,? 
4 (n—2) A, Ay>3 (n—3) A? 
5 (n—3) A, A; >4 (n—4) A,? 
n(n—[n—2]) Anis A, Sire 1) (n—[n—1]) A2,_, 
or 2n A, 5A, >(n—1) A®,_}. 
And if either of these conditions have place, we may infer 
the existence of imaginary roots in the proposed equation. 
Hence if we call any term in an equation, which lies between 
two terms with like signs, the middle term, we have the fol- 
lowing general principle, viz. 
If the product of the first and third of the three terms, mul- 
tiplied by the exponent of the first and by ” minus the expo- 
nent of the third, be greater than the square of the middle 
term multiplied by the exponent of that term and by x minus 
the same exponent, the equation must have imaginary roots. 
The well-known principle of De Gua is obviously included 
in this rule. 
Each one of the series of inequalities given above involves 
three of the given coefficients; and, as in the cases at first 
considered, if either of these sets of three be preserved, it mat- 
ters not how the remaining coefficients be altered: it follows 
therefore that the preceding conditions are perfectly inde- 
