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3. That the Coefficient of the third Term is dedu- 
ced from the Coefficient of the^fecond Term, by multi, 
plying after the fame manner, each of its Parts by the 
Index of e and dividing by 2 e. In general, the Co- 
efficient of any Termj/Ms deduced from the Coeffici- 
ent of the proceeding Term, that is ofj/*'~’ ^ by multiply- 
ing every Part of that Coefficient by the Index of e in 
it, and dividing the Produd by r e. 
Lemma XII. If you fubftitute any two Chiantities 
K and L in the Place of x in ^ A x * + B a;* — 
Cac’ + D, and the Quantities that refuk from thefe 
Subftitutions be affeded with contrary Signs, the 
Qjiantities K and L muft be Limits of one or more 
real Roots of the Equation Bac'- — Cw 
+ D=o. That is, one of thefe Quantities mud be 
greater, and the other lefs than one or more Roots of 
that Equation. 
For if you fuppofe that a^h^c^d, are the Roots of 
that Equation, then it is plain from the Genejis of 
Equations, that x^ — AAf* + BA;* — Gx + D = 
X — a X X — h X X — c X A’— - and therefore 
K and L being fubftituted for aa in at — axx — b x 
X — c y, X — dy the Produd becomes in the one Cafe 
pofitive, and in the other negative; fo that one of the 
Fadors x — at — Z?, a; — c, x — d muft have a 
Sign when K is fubftituted for x in it, contrary to the 
Sign which it is affeded with when L is fubftituted in 
in it for x, fuppofe that Fador to be a; — b and 
fince K — h and L — Z? are Quantities whereof the 
one is pofitive, and the other negative, it is manifeft 
that h one of the Roots of the Equation muft be lefs 
than one, and greater than the other of the two Qjjan- 
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