932] 
ON SYMMETRIC EUNCTIONS AND SEMINY ART ANTS. 
299 
is the diminished value of S", and finally making the numerical change, if T be 
c cL 
what S'" becomes on writing therein b, ... for b, c, d, this gives 
(db + 2 bd c + ...) T = 0, 
viz. T is a seminvariant. 
49. Assume that, for the weights up to a certain weight w, the forms of the 
sharp seminvariants are known: and for the weight w consider a seminvariant I(ca)F: 
here if I be given, the first theorem establishes a limit F' such that is in iO 
not higher than F'. For instance, when w = 12, if I = dj, the coefficient of j as being 
a seminvariant can only be d oo b 3 , and thus the seminvariant contains a term b 3 j, or 
the final term F must be in AO not higher than b 3 j; the degree is thus = 4 at least. 
Similarly, if F be given, then the second theorem determines a limit I' such 
that I is in GO not lower than /'. Thus when w = 12, as before, if F=b i cd 2 , then 
diminishing the letters we have be 2 , a term belonging to /oo&c 2 ; the diminished form 
has thus terms a 4 (a 2 ./, be 2 ), so that augmenting these the seminvariant has terms 
b 4 (b 2 g, cd 2 ) and thus the initial term I is in CO not lower than b s g. 
50. A limit for I or F, when the other is given, can also in some cases be 
found as follows: Considering a seminvariant of the weight w as before, and denoting 
its extent and degree by a and 8 respectively, then we have a8 — 2w — 0 or positive; 
that is, cr8 = 2w at least; here given I, we have cr, and then 8 = — at least; and 
(7 
given F we have 8, and then o- = ~k- at least. 
o 
51. We may now explain the MacMahon linkage; for a given weight, we write 
down in two columns the initials or non-unitaries in CO, and the finals or power- 
enders in AO: by what precedes, it appears that we cannot combine the terms of 
the one column each with the term opposite to it in the other column; what we 
do is: beginning with the top of the column of initials, we combine successively 
each term with the highest admissible term in the column of finals: or beginning 
with the bottom of the column of finals, we combine successively each term with 
the lowest admissible term in the column of initials.