( 68 ) 
Cor. I. Suppofe we are to compare E'E' the Sum 
of the Squares of the Parts of E with D'F^ the Sum 
of the Products of the fimilar Parts of D and F ; in this 
Cafe s vanifhes, and therefore — r xr E' E' muft ex- 
ceed X D' F'. Let n — r x r = m and confequently 
n — r — ixr — i — m — n \ n r — 2X 
r — 2 = m ^ n r — 3X?^ — 3 = 
m — 3 ;/ + 9 5 ;; — r — 4 x r — = m — 4?/ + 1 6. 
Since it is plain that n — r- — q xr — q =n r xr 
— Then by this Propofition, fuppofing 
mx^^ — xD'F'= 
m — n \ X D'F' — -ixCG' = h’ 
m — x;/ + 4 X C'G' — 3 oB'H' = ^' 
m — 3^/ + 9 X B' H' — 5'6A'F =z df 
m — 4;/+i6x A'F — 90 K' = 
The Quantities always pofitive 
when the Roots of the Equation are real Quantities alfed- 
ed with the fame Sign. The Coefficients prefixed to the 
negative Parts are the Numbers 2 ,ii,30j5'6.^9o, whofe 
Differences equally increafe by the fame Number 8. 
C o Fv. II. Suppofing as before, that n — r x r = '^ j 
and alfo that w-xw< — n 1 z= m! x^n — x ;/ + 4 
=z m" j X ^ — 3 + 9 = m'" 6cc. it may be de- 
•monftrated after the manner of this Propofition, that if 
WE'E'— xxi 2 CG' = ^" 
E'E' — X X IX X 30 B' H' = 
id” E'E' — X X 11x30 X 56 A'F = a!'” ^^c, 
Then 
