( 524 ) 
of all the Pairs of Roots or = a b + ^ c -f* a d + 
af * 4 - & g -f* &c. F> the Sum of the Prod lifts of all 
the Ternaryes of Roots or — abc-^abd-^abe-^ 
abf+abg — I - & o • F — ci b c d' — J— a b c 6 —J” ci b cf “|“ 
abeg-\- &c. F = a b c d e -j- a b c d f -(- ab c dg + 
bcdef -j- &c. and fo on. Let (as in this Propoli- 
tion) M reprefent any of thefe Coefficients, L, N the 
adjacent Coefficients, and m the Exponent of M ; let 
Z reprefent the Sum of the Squares of all the poffible 
Differences between the Terms of the Coefficient 
let a, be the Sum of all thofe of the forefaid Squares 
whofe Terms differ by one Letter, /g the Sum of all 
thofe Squares whofe Terms differ by two Letters, y 
the Sum of thofe Squares whofe Terms differ by- 
three Letters, £ the Sum of thofe Squares whofe 
Terms differ by four Letters and fo on. Thus if 
J\j[ — F = a b c d € — J— ci b c d — j-* ci b c d g — 
then Z— ab c de — abc df\ 2 -\-abc de — ab cdg\ 2 -{- 
a b c d e a b c fg\ 2 + b c d e f — a b f g h\ 2 + &c. 
a — ci b~cd e~ cTb c df \ 2 -j- a b c d e — a b c d g\*-\- 
abode — ab c dh \ 2 + b c d ef — b c d e g\* -\- &c. 
0 — abc de — ab c f g\‘ ab c d e — abcfh\ ‘ + 
bcde f—acdfh\‘+ f$c. y = abcde — abfg h\‘- f- 
abcdf—abTgh\* + = abc de — afghk\ + 
ac dfg — abe h k\- + &e. This being laid down I 
fay that the Square of any Coefficient ft 1 multiply d 
by the Fraction 
m x n 
m 
__ ~ exceeds the Rect- 
m-\- i%n — m + i 
angle under the adjacent Coefficients L x N by 
n + i X Z 
