titles K nnd L : So that K and L rauft be the Limits 
of the Root h, * „ 
I fay farther, that the Root, whereof K and L are 
Limits^ inuH: be a real Root of the Equation * for the 
Produd of the Fadors that involve impoffible Roots in 
an Equation can never have its Signs changed by fub- 
ftituting any real QiJantity whatfoever in place of 
becaufe the Number of fuch Roots is always an even 
Number, and the Pr odud of any two of thefe Roots 
fuch as X ' M — . “s/ — fi^ and AT — -j- — fi is 
X m\ n which mufl be always pofitive, what- 
ever Quantity^ be fubftituted for at while ;/ remains 
poiitive, that is, while thefe two Roots are impoffible. 
Lemma Xlll. If you fubftitute K and L for x in 
AC*” K X ^ <3cc. and the Quantities 
that refult be affeded with contrary Signs, then ffiall 
K and L be the Limits of one or more real Roots of 
the_ Equation a^« — Aa;”““* + B SiC. = o.. 
This may be demonftrated after the fame Manner as 
the laft Lemma. 
T H EO R E M I. If are the Roots of the 
Equation x^— Ax> -f — C*- + D = o.they. 
fhall be the Limits of the Roots of the Equation 
^ X ^ 3 A AC’ ^ — |- 2, B AT — — C = o. 
Suppofe ^ to be the lead Root of the biquadraticlc 
^ ^ Aac’^-FBat* — CAr-j-D==o, h the fecond 
Root, c the third, and d the fourth, and the Values of 
y in the Equation in the Lemma, will be ^ 
b •^eyC — e, d — e then by fubftituting fucceffive- 
Aya,hyC,d for e in that Equation of;;, one of the 
Values of y will vaniffi in every Subftitution, and the 
firft Term of the Equation of j;, vl^. e * Ae^ + 
= + D vaniffiing, the Equation will be re- 
duced to a Cubick of this Form. a p 
