932] 
ON SYMMETRIC FUNCTIONS AND SEMINVARIANTS. 
301 
Or beginning at the bottom of the column of finals, 6 12 is linked with c 6 , that 
is, we have (c 6 oo b 12 ), b 8 c 2 with c 3 d 2 , that is, we have (c 3 d 2 oo b 8 c 2 ); b 6 c 3 cannot be 
linked with d 4 , for the initial must be in CO not lower than b s e, but it is linked 
with the lowest term c 4 e for which this condition is satisfied, that is, we have 
(c 4 e oo b 6 c 3 ); and so on. 
The Unibral Notation. Strok’s Theory. Art. Nos. 53 to 56. 
53. Employing the umbras a, /3, 7, 8, ..., which are such that 
a = /3 = 7, ..., =b; a 2 =/3 2 = 7 2 , ..., =c; a 3 = /3 3 = y 3 , =d; 
and so on, then for instance 
(a — /3) 2 = a 2 — 2a/3 + /3 2 , =c — 2b 2 +c, = 2 (c — b 2 ), 
a seminvariant; 
(a — ¡3) 2 (a — 7) = a 3 — 2a 2 /3 + a/3 2 — a 2 <y + 2a/3y — ¡3 2 y, 
— d — 2 be + bc — bc + 2 b 3 — be, = d — Sbc + 2b 3 , 
a seminvariant: and so in general any rational and integral function of the differences 
of the umbrae developed and interpreted is a seminvariant. For the seminvariants of 
a given weight, e.g. w — 6, Dr Stroh * considers the function 
il 6 = (ax + /3y + yz + 8w + et + £u) 6 , 
where x, y, z, w, t, u are numbers the sum of which is =0, or we may if we 
please have more than 6 such numbers: the expression is obviously a function of 
the differences of the umbrae and it is thus a seminvariant. To develop its value, 
observe that after expansion of the sixth power we have sets of similar terms, for 
instance a 6 x 6 + ¡3 6 y 6 + ... which putting therein a 6 = /3 6 = 7®, ... = g become = g. Sx 6 , and 
generally each set becomes equal to a literal term multiplied by a symmetric function 
of the x, y, z, w, ...; introducing capital letters to denote the elementary symmetric 
functions of these quantities, and recollecting that their sum is assumed to be = 0, 
say we have 
1 + Cs 2 + Da 3 + Eh 4 + ... = 1 — x&. 1 — ys . 1 — zs , 
(that is, 0 = Sx, + C = Sxy, — D = Sxyz, &c.) then by aid of the Table VI (b) writing 
therein 0, C, D, E, F, G for b, c, d, e, f, g, we find 
il 6 = (ax + fty + 7z .. .) 6 = a 6 Sx 6 + Qa 5 SSafy + &c., 
* See the paper “ Ueber die Symbolische Darstellung der Grundsyzyganten einer binären Form sechster 
Ordnung und eine Erweiterung der Symbolik von Clebsch,” Math. Ann. t. xxxvi. (1890), pp. 262—303, in 
particular § 10, Das Formensystem einer Form unbegrenzt hoher Ordnung.