443 
1888 - 89 .] Dr T. Muir on the Theory of Determinants. 
that are affected, the like permutation being given to the superior 
indices. Making the choice of the superior indices of the c’s, let us 
permute them in every possible way, and to each term thus derived 
from cc-[cf . . . prefix the sign + or — according as its superior 
indices constitute a positive or negative permutation. By so 
doing the left-hand side of our identity becomes 2 ± ce{cf . . . c^> 
and, owing to the consequent permutation of the superior indices of 
the as, each term on the right-hand side gives rise to 1 . 2 . 3 . . .(n + 1) 
terms whose signs are the same as the signs of the terms correspond- 
ing to them on the left hand side ; — in other words, each term 
a a 'a" . . . a (n) . a a 'a'/ . . . dP gives rise to the compound term 
r s t w v s t w ° *■ 
«AV • • ^ • 2 ± a r°.' a /' ••••“' 
(n) 
We thus reach the result 
2 ± cc/c . . . c (n) == S (a a 'a” . . . a'f . 2 ± a a 'a" . . 
— l z n \ r s t n — r s t 
(n)\ 
* ) 
Although the number of terms on the right is the same as before, 
viz. (^9 + l) w+1 , arising from giving to each of the n + 1 indices 
r, s, t, . . . , vj any one of the p + 1 values 0, 1, 2, . . . , p, it 
has now to be noticed that a goodly proportion of them must vanish 
because of the fact that 2 + a a 'a" . . . a ^ = 0 when any two of 
its inferior indices are alike. The right-hand side will thus not be 
altered in substance if the summatory symbol be now taken to mean 
that r, 8, t, ... , w are to be any n + 1 of the p + 1 indices 
0, 1, 2, . . . , p. If p be less than n it will be impossible to 
have r, s, t, ... t w all different, so that in that case the right-hand 
side must be 0. This is Jacobi’s first proposition, and it constitutes 
his addition to the multiplication- theorem. His formal enunciation 
of it is (p. 309) : — 
Sit 
4 W*V*> + + . . . + a 
quotiesjp<% evanescit determinans 
2 ± cc x V 2 
S n ) » 
(xvm. 6) 
