1898 - 99 .] Dr Muir on a Single Term of a Determinant. 475 
EVEN CIRCULAR SUBSTITUTIONS. 
(47) The last of the five rules of signs may be quickly disposed 
of. Like Cauchy’s it makes use of “ circular substitutions,” but, 
unlike Cauchy’s, is not explicitly dependent upon the number of 
elements in the permutation. The easiest way, therefore, to con- 
nect it with the others is to deduce it directly from Cauchy’s. 
Denoting then by K e the number of “ circular substitutions ” with 
an even number of elements in each, and by k 0 the number of “ cir- 
cular substitutions ” with an odd number of elements in each, so 
that 
K e + K(, = K , 
we see that the total number of elements in the K e substitutions is 
even, and the total number in the k 0 substitutions is even or odd 
according as k 0 is even or odd. From these it follows by addition 
that the total number of elements in both kinds of substitutions — 
that is, n — is even or odd according as k 0 is even or odd, and there- 
fore that 
n — k 0 is even. 
But by Cauchy’s rule the sign of the permutation is 
/ +« ) 
{- ) e 
, >n — k , . — k 
e., (-) ».(-) *> 
e -> (-•) 
which is Jenkins’ rule. 
