447 
1888-89.] Dr T. Muir on the Theory of Determinants. 
being in question, and comparison between it and the primitive 
permutation, 
a, b, c, d, e, f g, 
having been instituted, we are directed to form the members 
(“ termes ”) of the permutation into groups, commencing to form a 
group with e and a , because they occupy like positions in the two 
permutations, putting b in the same group because it occupies the 
same position in the second permutation as one already in the group 
occupies in the first permutation, putting c in for the same reason, 
making d constitute a group by itself, and finally putting / and g 
together to form a third group. We are directed further, to write 
the members of each group in such an order that any member and 
the one following it may be found to occupy like positions in the 
primitive and derived permutations respectively. The result thus is 
(a, e, c, b ), (d), (/, g ) , 
or (e, c, b, a), (d), (y, /) , 
it being possible to write the first group in four ways, and the last 
in two. Now all this is nothing more than an unreasoning way of 
arriving at the circular substitutions which are necessary for the 
derivation of the given permutation from the primitive one. 
Cauchy himself, indeed, in pointing out that there would only be 
one way of writing a group if the members were disposed in a cir- 
cumference instead of in a straight line, says: — “C’est par ce motif 
que dans le tome x du Journal de VJ^cole Polytechnigue j’ai designe 
sous le nom de substitution circulaire l’operation qui embrasse 
le systeme entier des remplacements indiques par un meme 
groupe.” It must be borne in mind, however, that not only the 
operation, but the symbol of the operation, was so denoted, 
and such being the case, we may then very pertinently ask, What 
is a group in Cauchy’s usage but the symbol of a circular sub- 
stitution ? 
The peculiarity of using the number of groups to separate the 
various permutations of a, b, c, d, . . . . into two classes makes its 
appearance in the following sentence (p. 147): — 
