1887 .] Dr T. Muir on the Theory of Determinants. 463 
number of elements between A and B in value and situated to the 
right of B; then finding in like manner the sum of the place- 
indices for the new permutation; and finally comparing the two 
sums. The concluding sentence is as follows : — 
“ Denn da so ist die Summe der Stellenex- 
ponenten der . zweyten Permutation um cl + e+1 -e + d oder 
um 2^+1 grosser, als bey der ersten Permutation ; folglich 
gilt das auch bey der Summe der um 1 verminderten Stellen- 
exponenten, da bey beyden Permutationen r einerley ist. 
Also ist die eine Summe gerade, die andere ungerade, folglich 
haben nach (4) beyde Permutationen verschiedene Zeichen.” 
As immediate deductions from this, it is pointed out that 
The sign of any one permutation may be determined tchen the 
sign of any other is known , by counting the number of interchanges 
necessary to transform the one permutation into the other ; (iii. 8.) 
and that 
If one element of a permutation be made to take up a new place , 
by being , as it were, passed over m other elements, the sign of the 
new permutation is the same as, or different from , that of the original 
according as m is even or odd. (iii. 9.) 
A third corollary is given, but it is, strictly speaking, a self- 
evident corollary to the second corollary, and is quite unimportant. 
Bothe’s next theorem is — 
The permutations of 1, 2, 3, . . . . , n being arranged after the 
manner in ivhich numbers are arranged in ascending order of magni- 
tude, any tiro consecutive permutations will have the same sign, if the 
first place in which they differ be the (4n + 3) th or (4n + 4) th from the 
end, and will be of opposite sign if the said place be the (4n+ l) th 
or (4n + 2) th from the end. (iii. 10.) 
Thus if the permutations of 1, 2, 3, . . . . , 10 be taken, and 
arranged as specified, two which will occur consecutively are 
8, 4, 9, 3, 10, 7, 6, 5, 2, 1 
8, 4, 9, 5, 1, 2, 3, 6, 7, 10; 
and as the first place in which these differ is the 7 th from the end, 
it is affirmed that the signs preceding them must be alike. The 
VOL. xiv. 31/1/88 2 G 
