925] on wahing’s formula. 215 
the numerical coefficient of 
b m ~ e c c d d e e ... (c + d + e + ... = 0) 
being 
, N C+e + e+ . Km. m — (6 — 8 + 1). m — (6 — 8 + 2)... m - (6 — 1) 
( ' S lie. lid. lie ... (II2) C (I13) d (I14) e ... ‘ 
It is convenient to write down the literal terms in alphabetical order (AO), 
calculating and affixing to each term the proper numerical coefficient; thus taking 
1 + ba + Czj^2 + ••• =(1 —®b)(1 -/3x)(1 —<yx) 
we find 
- 120$ 6 = g . 1 
bf 6 
ce . - 15 
d 2 . - 10 
b 2 e . + 30 
bed . + 120 
c 3 . + 30 
b s d . - 120 
b 2 c 2 . - 270 
b 4 c . + 360 
b e . - 120 
+ 541 
this expression, as representing the value of the non-unitary function S 6 , being in 
fact a seminvariant. 
It is to be remarked that the foregoing expression for the sum of the mth 
powers of the roots of the equation 
x n + bx n ~ 1 + cx n ~ 2 + ... = 0 
ls > ln fact, the series’ for x m continued so far only as the exponent of b is not 
negative: see as to this Note XI. of Lagrange’s Équations Numériques. For the à 
posteriori verification, observe that we have 
7 c d 
x + b + — + — H -... = 0, 
or writing for a moment u = — b, say this is