298
ON SYMMETRIC FUNCTIONS AND SEMIN VARIANTS.
[932
47. As to the first of the foregoing inversions c 2 g, f 2 , it is proper to remark,
that filling up two compartments of the square we have
1
2
1
.
1
c 2 d 2 b 2 cd 2
where the meaning of the numbers (i, i) has to be considered: the first (i) seems
to indicate a seminvariant c 2 g oo c 2 d 2 , but there is in fact no such form, what it really
indicates is a form 0c 2 g + f 2 ao c 2 d 2 , that is, f 2 oo c 2 d 2 ; and similarly, the second (i)
seems to indicate a seminvariant f 2 oo b 2 cd 2 , but there is in fact no such form, what
it really indicates is f 2 oo c 2 d 2 + 0b 2 cd 2 , that is, f 2 oo c 2 d 2 . The explanation is correct,
but to make it perfectly clear some further developments would be required. The
like remarks apply to the inversion cdf, ce 2 .
The MacMahon Linkage. Art. Nos. 48 to 52.
48. We require the two theorems:
The first is: if a seminvariant S has q for its highest letter, then d q S is also
a seminvariant.
The second has presented itself for unitariants (ante No. 31); for seminvariants
the form is less simple, viz. If in any seminvariant, attending only to the terms of
the highest degree, we therein change b, c, d, e, ... into b, 2c, 6d, 24e, ... and then
diminish the letters (that is, replace each letter by the next preceding letter) and
in the result so obtained change b, c, d, e, ... into b, ~^^, ... we obtain a
seminvariant. For instance g — 6bf+15ce—10d 2 , in the terms of degree 2, making
the numerical change we have — 7206/+ 720ce — 360d 2 , and then diminishing the letters
and making the numerical change, we obtain —720 ^ + 720^ — 360^-, that is,
— 30 (e — 4bd + 3c 2 ),
a seminvariant.
For the proof, observe that the equation AS = 0, attending therein only to the
terms of the highest degree, gives (26d c + Seda + ...)S'= 0, if S' denote the terms of
the highest degree: making the numerical change, this is (bd c + cdd+ ...)S", if S" is
what S' becomes thereby; diminishing the letters, this is (0& + bd c + ...)S'" = 0, if S'
