_ ( 9.1 ) 
6Cc. = o, beginning with the fecond : Then if the 
Square of any Term multiplied by the Fradion over 
, it be found lefs than the Produd of the adjacent Terms, 
fome of the Roots of that Equation muft be imaginary 
Quantities. There remain many things that might be 
added on this Subjed, but I am afraid you will think 
I have faid as much of it as it deferves ; and therefore 
I (hall only add the Demonftration of fome Algebraick 
Rules and Theorems that are very eafily deduced from 
the xi^^ Lemma. 
I. The Rule for difcovering when two or more Roots 
of an Equation are equal, immediately follows from 
that Lemma, Suppofethat two Roots of the Equation 
AT" — 6Cc. = o are 
equalj and two Values of jv (which is equal always to 
X — ^)wiil be equal, Suppofethat^ is equal to one ofthofe 
two equal Values of x ; and two Values of j/ will va- 
nifli,and confequently y ^ muft enter each of the Terms 
of the Equation of y ; and therefore in this Cafe the 
firft and fecond Term of the Equation of y in Lemma 
muft vanifti, that is, e” — 
Ce 6cc. z= ,o and — i X A^”~ ^ + 
— XX 3 xC^"“'^ 6cc. = o at the 
fame time ; and confequently thefe two Equations muft 
have one Root common, which muft be one of thofe 
Values of x that were fuppofed equal to each other. It is 
manifeft therefore that v/hen two Values of x are equal 
in the Equation x^ — A + B.v”“'‘<?Cc.= o, 
one of them muft be a Root of the Equation 
— I X A^”““* + X Ba’”"“' 6cc. = o. 
If three Values of Arbefuppofed equal amongft them- 
felves and to then three Values of j/ (= a’ — e) will 
vanifli, and the firft three Terms of the Equation of y 
in 
