1888-89.] Dr T. Muir on the Theory of Determinants. 749 
even or odd according as the two permutations belong to the same 
or different classes; for, by the above theorem, every interchange 
makes only one group more or one group less, and consequently the 
total number of interchanges, and the net increase or diminution of 
the number of groups, must he both even or both odd. The 
counting of interchanges may thus he substituted for the counting 
of cycles. 
Finally, Cramer’s rule is introduced, in which, as we know, it is 
neither cycles nor interchanges that are counted, hut inverted pairs, or, 
as Cauchy, like Gergonne, calls them, inversions. To establish the 
rule, it is clear that two courses were open, viz., to connect inversions 
directly with cycles or to connect them with interchanges. The 
latter course is taken, the requisite connecting theorem being that the 
interchange of two elements of a permutation increases or diminishes 
the number of inversions by an odd number , an odd number of 
interchanges thus corresponding to an odd number of inversions, 
and an even to an even. The proof is not direct, like Rothe’s, 
being effected with the help of a fourth related entity, the difference- 
product. The order of thought in it is as follows : — If we define 
the difference-product of the primitive permutation a, b, c, d, .. . 
to he 
(a - b)(a - c) . . . . (b-c) . . . . , 
then it is clear that in the difference-product of any derived permu- 
tation there will he found exactly as many factors with changed 
sign as there are inversions of order in the permutation. A change 
of sign in the difference-product thus becomes a test for the existence 
of an odd number of inversions, and consequently, instead of the 
theorem just enunciated, it will suffice to show that the interchange of 
two elements of a permutation alters the sign of the difference-product. 
This Cauchy says must be true, for, the elements being h and k, it 
is manifest that the factor which involves them both, 
h-k or k - h, 
must change sign, hut that the factors which involve them and any 
third element s constitute a partial product 
(h - s)(k - s) or ( h - s)(s - k), 
the sign of which cannot change. 
(hi. 37) 
