the Imaginary Roots of Equations. 255 
tially developed roots, when an imaginary pair is indicated, 
provided f,, (a) has approximated closely to zero. 
Moreover, when three roots concur in several leading figures, 
we should in like manner arrive at a cubic indicator; and 
when there are four such roots, at an indicator of the fourth 
degree; andsoon. And these indicators, like as in the qua- 
dratic, would point out the initial directions which the sepa- 
rated roots take. But it is unnecessary to examine these in- 
dicators of the higher orders, all of which are ultimately re- 
ducible to quadratics: so that in examining minute intervals, 
in the theory of equations, like as in discussing the elements 
of a curve surface, the quadratic indicator is sufficient to sup- 
ply, in both cases, all the desired information. 
I shall now return to the general criteria at first given, and 
shall show their importance in detecting imaginary roots, pre- 
viously to any actual development, and solely from an exami- 
nation of the proposed coefficients ; and shall thence deduce 
the rules of Newton before adverted to. 
It is well known that for the purpose of examining into the 
character of the roots of an equation, as to real and imaginary, 
we may replace that equation by a series of limiting equations 
of inferior degree: as for instance, if the equation be above 
the third degree, by a series of cubic equations; or, if we 
please, by a series of quadratics. In the present inquiry it 
will be proper to take the limiting cubics, and not the qua- 
dratics, as Maclaurin, and all other investigators of Newton’s 
rule have, I believe, done. 
If any of these limiting cubics indicate imaginary roots, 
such indications will of course also imply imaginary roots in 
the proposed equation. But several indications, apparently 
distinct, may offer themselves in these equations, which upon 
closer examination may be found to be necessarily dependent, 
and thus to concur in pointing to but a single imaginary pair. 
Distinct imaginary pairs can be inferred only from distinct 
independent conditions: we have therefore to inquire how 
these are to be discovered in the series of limiting equations 
alluded to. 
1. And first we may remark that a cubic equation consists 
of only four terms; and as but one imaginary pair can enter 
into it, it follows that whether the criterion at page 186 hold 
with respect to the three leading terms, or with respect to the 
three final terms, or in reference to both sets of three, one ima- 
ginary pair, and one only, is implied. 
2. The cubics we are considering are so connected together, 
that if the criterion hold, or fail, in reference to the three 
leading terms of one, it must of necessity, in like manner, hold, 
