1899—1900.] Dr Muir on the Theory of Alternants. 115 
it being pointed out that any term of the expansion is got from 
the preceding by cyclical permutation of the suffixes, and that the 
signs are all + when the number of elements is odd, and alter- 
nately + and — when the number of elements is even. The case 
of Laplace’s expansion-theorem, which is here used, is easily seen 
to be that where the orders of the minors are n- 1 and 1. Thus 
using later notation, we have 
= |6W| - | a l c 2 d*\ + | a}b 2 d*\ - |aW| , 
= bcd\b°e l d 2 \ - acd\a®c l d 2 \ + abd\aWd 2 \ - abc\a% l e 2 \ , 
which is the desired result. 
In connection with this, it is perhaps worth noting that the 
iresult being, by the same case of Laplace’s theorem, also equal to 
II a a 2 bed 
j 1 b b 2 cda 
1 c c 2 dab 
1 d d 2 abc , 
we may view J acobi’s first theorem as being equivalent to one of 
later date, viz. — 
$(abcd) = 
1 a a 2 a 6 
1 b b 2 5 3 
1 c c 2 c 3 
1 d d 2 d 3 
P( a l a 2 a S ■ ■ 
,.a n ) = (-)"- 1 
1 
a Y 
2 
• 
n-2 
. . 
CLc)CLo(Xt^ • • 
. a n 
1 
a 2 
2 
«2 ‘ 
n- 2 
. . a 2 
a x a 3 a 4 . . , 
. an 
1 
a n 
2 
a n * 
n-2 
• • a n 
a^a 3 . . . 
an- 
When the determinant is of even order, it is possible to use that 
■case of Laplace’s expansion-theorem in which all the minors are of 
the 2 nd order. Thus 
