391 
1888-89.] Dr T. Muir on the Theory of Determinants. 
tion that a permutation and any other derivable from it by the 
simple interchange of two indices must, according to Cramer’s rule, 
differ in sign. This proposition is therefore attacked. The permu- 
tation 
(*). (l).+n A 
is taken in which the inferior indices are in their natural order 
1, 2, 3, . . . , n, and k and 1 being interchanged, there arises the 
permutation 
(i)« (*).+/. B 
The part preceding ( k) a in A is called I., which thus of course also 
denotes the part preceding (l) a in B : the part between ( k) a and 
(l) a +/3 in A or between (l) a and (k) a +P in B is called II.; and the 
remaining part common to both A and B is called III. The 
number of inversions in both, when 1 and k are left out of account, 
is denoted by y : the number in both due to k and the division III. 
is denoted by A : the number in A due to k and the division II. 
by A' : and the number in both due to the division I. and k by A". 
The counting of the inversions then takes place for the two permuta- 
tions. In the case of A there are the inversions due 
(1) to I. and k , which are X" in number. 
(2) to I. and II. 
(3) to I. and 1, . . . . a - 1 . . . 
(4) to I. and III. 
(5) to k and II., .... A' . . . 
(6) to k and 1, . . . . 1 . . . 
(7) to k and III., .... A . . . 
(8) to II. and 1, . . . . /5 - 1 . . . 
(9) to II. and III. 
(10) to 1 and III., ... 0 . . . 
and as those not counted here are y in number, the total is seen to be 
a + /? + y + A + A' + A"-l. 
Similarly in the case of B the total is found to be 
a + (3 + y + X- X + X" — 2 . 
