392 Proceedings of Royal Society of Edinburgh. [sess. 
But the former total exceeds the latter by 2A' + 1, and this being 
an odd number, the proposition is proved. (hi. 26) 
Before proceeding further it is important to note that Grunert 
here establishes a more definite theorem than he proposed to 
himself, viz., the theorem of Rothe (III. 7). If he attains greater 
simplicity it is in part due to the fact that instead of taking any two 
indices for interchange, k and r say, he takes k and 1. 
To prove now that the function constructed in accordance with 
Cramer’s rule will satisfy the requisite conditions, it suffices to show 
by means of this theorem that on making any one of the n— 1 
specified sets of substitutions the function will be transformed into 
one consisting of pairs of terms which annul each other ; in other 
words, to prove Vandermonde’s theorem regarding the effect of 
making- two indices alike. This is done; and then it is shown 
how x K can be got by interchanging x K and x l in all the given 
equations, the first step being of course to establish the fact that 
the denominator of x K and the denominator of x x only differ in sign. 
Bezout’s rule of 1764 is next taken up, and shown to be identical 
in effect with Cramer’s. The proof, by reason of the recurring 
character of the former, is inductive ; that is to say, it is demon- 
strated that, if the two rules agree in the case of n unknowns, they 
must also agree in the case of n+ 1. Paraphrasing the proof, but 
taking for shortness’ sake the case where n— 4, we say that it is 
agreed that both rules give in this case the signed permutations 
1234, - 1243, +1423, -4123, - 1324, +. . . 
Now for the case where n — 5 Bezout’s rule directs that to the end 
of each of these permutations, e.g., the permutation - 4123, a 5 is 
to be put, and asserts that the result - 41235 will be one of the 
desired permutations with its proper sign. That it is a permuta- 
tion of the first five integers is manifest, and since the number 
of inversions in 41235 is necessarily the same as the number in 
4123, its sign is correct according to Cramer’s rule. In order to 
obtain four other permutations, Bezout’s rule then proceeds to bid 
us shift the 5 one place and alter the sign, shift the 5 another 
place and alter the sign again, and so on. The result is 
+ 41253, -41523, +45123, -54123. 
In regard to this, it is clear as before that permutations of the 
