1898 - 99 .] Dr Muir on a Single Term of a Determinant. 445 
und verschiedene, wenn er ungerade ist, weil man durch so 
viel Vertauschungen, als der Unterschied der Stellen betragt, 
aus der einen auf die andere kommen kann. So haben 
also die bey den Permutationen 
478 10 195263 
und 
48 10 1952763 
einerley Zeichen, weil bios das Element 7 bey der einen in 
der zweyten, bey der andern in der achten Stelle steht, denn 
man kann durch sechs Yertauscbungen aus der ersten Per- 
mutation die zweyte ableiten.” 
The question of priority, however, is of very little moment. 
(5) The fifth, and perhaps the only other rule, is of quite 
recent date, viz., 1895, having been given by Mr Morgan Jenkins 
in the Messenger of Mathematics , xxv. pp. 60-68. As in Cauchy’s 
case it is circular substitutions that are counted. It may be stated 
as follows: — If K e be the number of even circular substitutions 
necessary to transform the given permutation into the standard 
permutation , the sign of the given permutation is ( - ) K e. For 
example, the given permutation being 1736542 we decompose 
the substitution 
/I 7 3 6 5 4 2\ 
111 2 3 4 5 6 7/ 
into circular substitutions, neglecting all those which contain an 
odd number of elements, viz., 
and keeping count of those which contain an even number of 
elements ; and there being only 1 of the latter, viz., 
the sign is ( - ) 1 . 
(6) The main points connected with the five rules may be 
summed up as follows : — 
