304
ON SYMMETRIC FUNCTIONS AND SEMIN VARIANTS.
[932
55. We can, by means of the umbral notation, write down for the blunt sem-
invariants of a given weight (indefinite forms, not the above-mentioned specific forms)
expressions far more simple than those which are given by the foregoing theories:
we can, in fact, find without difficulty monomial umbral expressions; and in many
cases obtain also the sharp forms. To illustrate this, I consider the weight 10: I
write down for convenience the symbols of the sharp forms (though the knowledge
of these is in nowise required) and I form a table as follows:
Sharp forms,
finals in AO.
k 00 f 2
1 (a-/?) 10
ci „ ce 2
2 (a — fiy (a —y) 2
dh „ b 2 e 2
3 (a — /3) 8 (a — y) (a - 8)
eg „ bd 3
4 ( a -Pf (a-y) 3 (a-8)
f 2 „ o 2 d 2
5 (a — /3) 6 (a — y) 2 (a — 8) 2
c 2 g „ b 2 cd 2
6 (a-/?) 6 (a-y) 2 (a-S) (a - e)
ce 2 „ c 5
7 ( a - Z 3 ) 4 ( a - y) 2 ( a - 8 ) 2 ( a - e ) 2
cdf ,, b 4 d?
8 (a - /3) 6 (a - y) (a - S) (a - e) (a - £)
d 2 e „ b 2 c 4
9 (a - (3) 4 (a - y) 2 (a - S) 2 (a - e) (a - £)
c 3 e „ b 4 c 3
10 (a - A) 4 (a - y) 2 (a -8) (a - e) (a - £) (a - rj)
c 2 d 2 „ b 6 c 2
11 (a — f3) 4 (a — y) (a-8) (a - e) (a - £) (a - rj) (a - 6)
c 5 „ b w
12 (a — (3) 2 (a — y) (a — 8) (a — e) (a — l) (a - rj) (a — 6) (a
It will be observed that all the differences used are a —/3, a — y, ... containing each
of them an a; hence in all the forms we have a 10 , = k; in (a — /3) 10 , the lowest
term (in A 0) is a 5 /3 5 , =f 2 ; in (a — /3) 8 (a — y) 2 , the lowest term is a 4 /3 4 . y 2 , = ce 2 ; and
so on, viz. in each case the lowest term is the final term of the sharp form set
down in the same line.
56. The form (a — /3) 10 gives at once the sharp form k oo/ 2 ; we thus develop it:
a 10
a 9 /3
a 8 /3 2
<x 7 /3 3
a 6 /3 4
a 5 /3 5
/3 10
a/3 9
cd/3 8
a 3 /3 7
a 4 /3 6
1
10
+ 45
-120
+ 210
252
+ 1
10
+ 45
- 120
+ 210
= 2(k -
10 bj
+ 4 5ci
— 120dh
+ 210e^r —
126/ 2 );
— /3) 8 (a — y) 2 contains a term ot 10 , = k
it we employ the form (a — /3) 8 (a —
a 4 /3 4 . y 2 , = ce 2 , but there is no term
and thus gives a blunt form kaocd
y) (/3 — y), then here as before the
a 10 : there is a term a 9 /3, = bj,
