( 88 ) 
Theorem III. In general^ the Roots of theEqua* 
tlon X” — = o, 
are the -Limits of the Roots of the Equation nx^"^ ^ * 
— — <5cc. = o, 
or of any Equation that is deduced from it hy multi^ 
plying its ferms hy any Arithmetical ' Progrejfion 
l'^ d, I i.dyl+ ^d 6cc. and converfely the Roots 
of this new Equation will he the Limits of the Roots 
of the propofed Equation x^ — Aa;”"“* 
&c. = 0. 
This Theorem is demonftrated from the xi^^ and 
xiih^ Lemmata in the fame manner as the preceeding 
Theorems were demonftrated from the ^Lud xii^^ . 
From thefe Theorems it is eafy to infer all that is deli- 
vered by the Writers of Algebra on this Subjed. 
Theorem IV. Phe Equation a** — . A at”"" ‘ 
^c. = o will have as many ima- 
ginary Roots as the Equation nx'"’^^ -^n • — i x 
A<v”~^ — n — xxBx*”"^* &c. = o, or the Equa- 
tion Ex -xBx'”~'* -j- 3 Ca;”’“* 6cc. = o. 
Suppofe that any Root of the Equation nx"""^ ' — 
n — I xAa?"”**4- .ixBa?”~* 6cc. = o, as 
p becomes imaginary, and the two Roots of the Equa- 
tion A?** — A-v””’ + Bx* <5cc = o, which by 
Theorem III. ought to be its Limits^ cannot both be 
real Quantities ^ ' for it is manifeft from the Demonftra- 
tion of Theorem I. that if they are real Quantities, 
then being fubftituted for a; in ;/ a; ^ i x 
Aa;”"'^ + n — X xBaj’”’“* 6cc. the Quantities that 
refult. mult have contrary Signs, and confequently the 
Root /, whereof they are Limits^ muft be a real Root ; 
wrhich 
