the Imaginary Roots of Equations. 257 
when it fails: the alternations of sign, thus furnished, will de- 
note the number of imaginary roots, at least, which enter the 
equation: and this is Newton’s rule. 
The rule now established will be found a valuable adjunct 
in the modern theory of numerical equations; more especially 
in connexion with the researches of Fourier. We shall apply 
it to an example taken from the Analyse des Equations of this 
author :— 
P+ gt*+ 273 — 227+ 9r —-1=—0 
+ —-— + -—- = 4+ 
Hence the equation has four imaginary roots. In the work 
from which this example is taken, a good deal of labour is 
expended upon arriving at this conclusion. 
To what has now been done it may be proper to add, that 
although the criteria at page 187 have been deduced from 
Sturm’s theorem, yet they may be readily inferred from inde- 
pendent principles; and, moreover, without any direct re- 
ference to the limiting equations of Newton and Maclaurin, 
adyerted to above. For it is shown in the Theory of Equa- 
tions, p. 323, that if the general equation 
Bi a A, aE Ns Nes aaa oadies® Yee fer 
be transformed into another, by substituting «+r for x, then 
the third coefficient of the transformed equation, in order that 
the second may vanish, must be 
An. °“n(n—1) £ Aga V 
oer 2 ee 
Consequently if, when this evanescence takes place, the ex- 
pression here written have the same sign as A,, the zero, then 
occurring between like signs, will indicate imaginary roots. 
Hence, multiplying by the positive quantity 2 n A,2, two ima- 
ginary roots will be indicated provided we have the condition 
2n An.A, > (n—1) A®,_, 
or, by reversing the coefficients, provided we have the condi- 
tion 
2n A, A, > (n—1) A,? 
and this, applied to the final terms of the successive derived 
equations, will obviously furnish the series of criteria at p. 187. 
As imaginary roots are equally indicated though the third 
coefficient actually vanish along with the second—except all 
the roots are equal to r—it follows, as at p- 186, that the sign 
of inequality may be changed into that of equality without 
disturbing the indications. 
In terminating these investigations, I may perhaps be per- 
Phil. Mag. S. 3. Vol. 22. No. 145, April 1843, Ss 
