HISTORICAL SKETCH. 
7 
then making K = 5 
— i(c 2 «i + c 1 a 1 ) = —>S X 
and since = — a 4 c 2 = 7, 
making K = 4 
— t (c 3 a 2 + c 4 af + c 2 a\ + c^o) = - S 2 = — (a? — 2 a 2 ) 
and by equating the coefficients of the a’s, c 3 + c x = — 8 and c 2 + c 4 = 4, giving 
c 3 = 4 c 4 = — 1 
making /v = 3 
—| [ 2 c 5 a 3 + c 6 a 4 a 2 + ch (c 3 a 2 + c 4 af) + a 2 c 2 a 4 + cla] = s 4 s 2 — 2s 3 
from which are obtained 
c 5 3 c 6 — 1 
Finally c 7 is found to be 0, and 
[321] = — 12a 6 + 7 a b a x + 4a 4 a 2 — 3a 4 al — 3 a§ + a z a 2 a 4 
By the use of a determinant and symbolic multiplication, Brioschi 
also expressed Waring’s reduction formula (5) in a much simpler form 
than did its discoverer; thus 
where 
and in general 
^11^12 • 
• • U ln 
• ..*»] = 
R-21^22 • 
■ • U 2n 
(7) 
V'nlV'ni • 
■ • U nn 
'M' rs U 8r = 
[* S + *r] 
U rs U st . . . U vr = [x r + x 8 + • • • +*,] 
About the same time (1857) Cayley 5 republished the Hirsch tables, 
using the equation ( 1 ) and reversing the order of the coefficients in the 
tables, so that Hirsch’s principal diagonal became his sinister diagonal. 
He proved that each of its elements must be ( —1)”’, where w, called the 
weight, is defined for [x 4 x 2 . . . x„] by the equation 
W = x 1 + x 2 + ■ • • +x n (8) 
for example the weight of 1,a\a 2 a z is 7. 
Since if a\'a 2 - . . . a„ n , written (/fik 2 . . . k n ) is to have a coefficient 
other than 0 in the expression for [x 4 x 2 . . . x n ] in terms of the coeffi¬ 
cients, the relation 
^1 + ^2+ ‘ ' ‘ + X n — ^1 + 2k 2 + • • • 
must obtain, we may also calculate the weight from the equation 
w = / l 1 + 2k 2 + • • • + uk n 
5 A. Cayley: Philosophical Transactions of the Royal Society of London, vol. 147, 1857. 
