1898 - 99 .] Dr Muir on a Single Term of a Determinant. 471 
If a given permutation of n elements have k cycles of substitution , 
it can be transformed into the standard permutation by means of 
n — k interchanges , provided every pair of interchanged elements be , 
before interchange , in the same cycle. 
The equivalence of Cauchy’s rule of “ circular substitutions ” with 
the rule of “interchanges” is thus made manifest. 
(39) If in transforming a given permutation into the standard 
permutation we confine ourselves to effective interchanges only , the 
number necessary will still be n - k. 
For, an effective interchange, being an interchange which brings 
at least one of the two elements concerned into its standard place, 
must be an interchange of two consecutive elements in a cycle, and 
the only difference will be that a monomial cycle will be split off 
by every operation. 
Here, in passing, it may be noted as self-evident that the order 
in which effective interchanges are made is immaterial. 
(40) The interchange of two elements in different partial circular 
substitutions destroys the circular character of both substitutions , but 
makes it possible to form one circular substitution out of the two. 
Let the given circular substitutions be so written that one of the 
two elements concerned occupies the last place in the substitution 
to which it belongs, and the other the first place ; and let the 
substitutions so 
written be 
• • 
. ., x,»A, w , a \ 
and 
\ • • 
\a,p,y, . 
. . ., x>»/w 
\a,(3,y, . 
• • •, x', /,<*>/ 
After the interchange these become 
. ., x,«A,w,a\ 
and 
• • ■ 
• •, x',/, w ', a '\ 
. . ., XiM/ 
\(0, /?',/, . 
• • •, x'l^V/: 
which are manifestly non-circular, the break in the chain of the 
first occurring when we try to leave the item if/- into-co. If, how- 
ever, at this stage we neglect for a little the next item, and move 
on to the second substitution, we find that to has to be changed 
into /3', j3' into y\ . . . . , to' into a. All that is then wanted to 
