314
ON SYMMETRIC FUNCTIONS AND SEMIN VARIANTS.
[932
The Strohian Theory Resumed: Application to Perpetuants. Art. Nos. 65 to 71.
65. We can by means hereof establish, in regard to the specific blunt semin-
variants, a general theory of reduction, or say a theory of the relations which exist
between the seminvariants of a given degree and the powers and products of semin-
variants of inferior degrees. To exhibit the form of these, it will be sufficient to
take O a sum of two parts, = O' + O", but the more general assumption is O a
sum of any number of parts, = O' + O" + Cl"' + .... Taking then 0 = 0' + Cl", where
for the II' and il" separately the sum of the (a, y, z, ...) is =0, suppose that to
the (0, C, D, E, ...) of il there correspond (0, G', D’, E’, ...) for il' and (0, C",
D", E", ...) for il". We have
G = G’ + G",
D = D' + D",
E=E'+ E" + C'C",
F = F' + F" + C'D" + C"D',
G = G'+G" + G'E" + G"E' + D’D",
the law of which is obvious.
66. We have, for instance,
il 4 = (O' + O") 4 , = il' 4 + 60' 2 0" 2 + il" 4 , (since O' = 0, O" = 0),
that is,
(iG' + C'J c 2 = C" 2 c 2 + 6CV. G"b 2 + C"V
+ (E' + E" + C'G")b i + EV +E'V
where, and in what follows, c 2 , 5 4 , 6 2 are for shortness written instead of [c 2 ], [№],
[b 2 ] to denote the specific blunt seminvariants ending in c 2 , 6 4 , b 2 respectively.
The terms in C' 2 , G" 2 , E', E" are identical on each side of the equation and
destroy each other: omitting these, we have only the terms in G'G" which must be
equivalent on the two sides of the equation, and comparing coefficients we find the
relation
2c 2 + 6 4 = 6.6 2 . b 2 ,
which of course means 2 [c 2 ] + [5 4 ] = 6 [6 2 ] [6 2 ], viz. this is
2 (2e — 8bd + 6c 2 ) + (—4e+ 16bd + 12c 2 — 486 2 c + 246 4 ) = 6 (— 2c + 26 2 ) 2 .
In like manner, for O 6 = (O' + O") 6 , we have
(O' + C") 3 . d 2
4- {IT + D") 2 . c 3
+ {C' + G") (E' + E" + G’G") . b 2 c 2
+ (G’ + G” + G’E" + G"E' + D'D"). b s