( 8o ) 
made by multiplying, any given Number of them by 
one another, inuft be a maximum when thefe Qjjanti- 
ties are equal. But the Sum of the- Squares, or of any 
pure Powers of thefe Qjaantities, is 2! minimum, when 
the Quantities are equal. ' - 
THEOREM. 
Suppofe — A x”'^ ^ C x ” ^ 
T) x*^^ — 6Cc. = 05 to he an Equation that 
has not allits Roots equal to one another : Let rex- 
prefs the Dimenfions of any Coefficient D, and let 
^ w- I n — ^ n 
I z=zny. X X - 
3 ' 
- 6Cc. taking as ma- 
ny Favors as there are Units in r \ then fiall x 
he always greater than D, if the Roots of the Equa^ 
tion are real Quantities a fedled^ith the fame Sign, 
This may be demonftrated from the proceeding Pro- 
politions: But to demonftrate it from the laft Lemma- 
ta, let us affume an Equation that has all its Roots 
equal to one another, and the Sum of all its Roots 
equal to A, the Sum of the Roots of the propofed Equa- 
I r 
tion. This Equation will be at A I = o, or 
/ 
I ‘A* 
x^ — A ^ + nx X — r ^ X 
X n 
n — I n — ^ A’ 
X X X oCc. = o and it r ex- 
z 3 n^ 
prefs the Dimenfions of the Coefficient of any Term 
of this Equation (or the Number of Terms which 
pre- 
