1888.] Dr Muir on a Class of Bimfh Alternants. 301 
- a [\h‘^c^d^\ - + \h^chl^\] 
+ h [\a‘^c^d^\ - \a}d^d^\ + \a^c^d^\] 
-c{|aWl - \adhH^\ + \a%^d^\) 
+ d[\a%^c^\ - \a?-'b^c^\ + \a%^c^\) . 
Of the twelve terms here the four with the indices 2, 4, 5 are 
equal to - 1 a}h^d^d^ |, the four with the indices 1, 4, 6 are equal to 
\ aMi^c'^d^ \., i.e. to 0, and the rest = -| |. Consequently the 
final result is, as before, 
— \a}h‘^d^d^\ + \a%^c^d^\ , 
(4) Much of the detail, however, of this second method may be 
omitted. In the first place, the expansion with which it opened is 
unnecessary, because the multiplications to be performed can be 
noted without expanding ; in the second place, the result of only 
one multiplication need be written out ; and lastly, the summation 
of the columns can be made quite mechanical. What we should 
therefore really do in practice would be as follows. Glancing at 
the first row of the given determinant, viz., the row 
1 a + + 
we should write down the first four indices 
0 12 1 
and place under the first three of them the indices of the terms 
of the symmetric function, viz., 
4 4 0 
4 0 4 
0 4 4. 
Addition of the first line of indices to each of the three following 
lines would then be performed, the results being 
4 5 2 1 
4 16 1 
0 5 6 1 
which are the indices of the desired final alternants 
\a%^c^d}-\ , \a%^c^d^\ , \a%^c^d^\ 
or 
or, say - A(1 2 4 5), 0 A(0 1 5 6). 
