1899— 1900. J Dr Muir on Jacobis Expansion. 
141 
(8) A similar definition of a determinant is at once suggested, 
viz., 
One term of the determinant j a 1 b 2 c 3 . . . z n [ is + a^Cg . . . z n : 
this is increased to two by the cyclical permutation of n - 1, n 
accompanied by change of sign : these two are increased to six (i.e. 
2x3) by the cyclical permutation of n — 2, n - 1 , n ivithout altera- 
tion of sign : then these six are increased to twenty -four (i.e. 
2x3x4, by the cyclical permutation of n - 3, n - 2, n— 1, n 
accompanied by change of sign : and so on. 
Thus, the first term of | a 1 & 2 c 3 cZ 4 | is 
■J - afi^cyd 4 , 
from which by cyclical permutation of 3, 4 we obtain another 
afo^c^d^ , 
then by cyclical permutation of 234 without change of sign we 
derive from the former 
+ a 1 5 3 c 4 c? 2 + a 1 6 4 c 2 ^ 3 , 
and from the latter 
- afbgc^d^ - a-fi^cfl^', 
and lastly by cyclical permutation of 1234 and change of sign there 
is derived from these six the remaining eighteen : 
As before, the total number of terms, viz., 1*2’3 . . . n, is 
brought very clearly into evidence. 
