Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., late sadlerian professor of pure mathematics in the University of Cambridge (Vol. 13)

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
	        
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