1901-2.] Professor W. H. Metzler on Alternants. 
241 
1 
a 
a 2 
A 
aA 
1 
b 
5 2 
B 
bB 
1 
c 
c 2 
C 
cC 
1 
d 
d 
D 
dJ) 
1 
e 
e 2 
E 
eE 
and similarly for the others. 
The object of this paper is to show that for determinants of 
order five no such identity as (A) exists; that is, that it is not 
possible to find values of a, /3, y, S, e, p, 77, 6 to satisfy this relation, 
nor is it possible to find sets of values of these exponents to 
satisfy the relation. 
.(B) | a 0 b l c 2 D eE | . | A 0 B 2 C 4 D 6 E 8 | 
= | a?V c 2 D 2 eE 2 | -2 | A® B“ G* D Y E J | 
± Ian 1 c 2 D 4 eE 4 | -S | A° B' C p D” E® | ; 
where, of course, the conditions a + /? + y + S=18, and e + p + 77 + 0 
= 14, demanded by homogeneity, must prevail. 
This done, it follows that no such identity exists for determinants 
of higher order ; for the relation (B) being an identity, it follows 
that the coefficients of the various powers of A, B, C, . . . . vanish 
independently. Now the coefficient of e 3 F 10 , for instance, in the 
case of determinants of order six, is the relation (B), and if (B) 
cannot exist, there is no relation for determinants of order six, and 
so none in general. 
Write (B) in the form 
(C) X | a° b 1 c 2 D eE | • | 02468 | 
= | a° & i c 2 D 2 eE 2 | {ft | 01278 | +ft | 01368 | 
+ ft | 02358 | +/* 5 | 01467 | 
| 02457 | +ft | 03456 | 
+ ft 0 | 013410 | +ft x | 01269 | 
-t-fts I 02349 | } 
+ | 01458 | 
+ fi 6 | 02367 | 
+ /x 9 |012510 | 
+ />i 12 | 01359 | 
+ | a 0 b 1 c 2 D 4 eE 4 | {v x | 01256 | +v 2 | 01346 | +v 3 j 02345 | 
+ v 4 | 01247 | +v 5 -| 01238 | } 
| a 0 & 4 c 2 D 6 eE 6 | • | 01234 | , 
PROC. ROY. SOC. EDIN. — VOL. XXIV. 
16 
