256 Prof. Young’s New Criteria for 
or fail, in reference to the three jimal terms of that next in 
order. 
Hence the condition holding for the three leading terms of 
one cubic necessitates its holding for the three final terms of 
the next, so that the concurrence implies but a single ima- 
ginary pair. By examining our series of cubics, with these 
principles before us, applying the test to each group of three 
terms in succession, we shall obviously be able to distinguish 
those conditions which are really distinct and independent, 
from those that are not, and therefore to infer so many di- 
stinct imaginary pairs. ; 
If the first set of three, that is the leading terms in the first 
cubic, satisfy the criterion, we immediately infer the existence 
of one imaginary pair; if the next set—the final terms of the 
same cubic—also satisfy it, the preceding condition merely 
recurs, and supplies no new information. In this case the 
following set of three—the leading terms of the next cubic— 
must furnish the same concurring condition, by the second 
principle above; and so on, till we arrive at a set of three for 
which the criterion fails, thus putting a stop to the series of 
concurring conditions, and preparing the way for new and in- 
dependent conditions. As soon as the criterion again holds, the 
condition, being altogether independent of the former, must 
imply another and distinct imaginary pair; and so on, to the 
end of the series. 
Now the criteria which we have here supposed to be ap- 
plied to the terms, taken three at a time, of the successive 
limiting cubics, supply one after another the very expressions 
that we have exhibited at page 187; the three final terms of 
one cubic always furnishing the same expressions as the 
three leading terms of the next, as noticed above. Conse- 
quently, without the formal intervention of the limiting cubics, 
which have merely been introduced into the reasoning for the 
purpose of tracing the dependent conditions, we may at once 
apply the criteria (page 187) to the coefficients of the pro- 
posed equation, observing that when the condition recurs, in 
proceeding from one set of three to the next, the recurrence 
is to be regarded merely as a repetition of the same indica- 
tion; that as soon as it fails, preparation is made for the oc- 
currence of a new indication, and so on. 
Hence the indications that are really independent, and con- 
sequently the number of imaginary roots inferrible from them, 
may be thus noted. Under the first and last terms of the 
proposed equation write the sign plus; then, taking each of 
the intermediate terms in succession for a middle term, write 
under it the sign minus when the criterion holds, and plus 
