446 Proceedings of Royal Society of Edinburgh. [sess. 
positions of the theory of determinants, was prompted by the 
appearance of Jacobi’s memoirs, and by the consequent conviction 
that the work of 1812 had begun to bear fruit. The first paper, 
called a “note,” is introductory, on the subject of signed permuta- 
tions; the three others, called “memoirs,” correspond to Jacobi’s, — 
the first of them to Jacobi’s third, the second to Jacobi’s first, and 
the third to Jacobi’s second. 
The note, although on so trite a subject as the division of permu- 
tations into positive and negative, is most interesting. Cauchy’s 
original stand-point with regard to the subject is so far unaltered that 
the rule of signs specially known by his name is made fundamental, 
and all others deduced from it. The explanations preparatory for 
the rule are, however, on the lines of his paper of 1840, that is to 
say, it is groups and not circular substitutions that are spoken of. 
The preference is a little difficult to justify; for notwithstanding 
Cauchy’s assertion that groups come naturally into evidence, the 
idea is far-fetched as compared with that of circular substitutions. 
He says (p. 145) : — 
“ Si l’on compare une quelconque des nouvelles suites * a 
la premiere, on se trouvera naturellement conduit par cette 
comparaison a distribuer les divers termes 
a, b 9 c, d ... . 
en plusieurs groupes, en faisant entrer deux termes dans un 
meme groupe, toutes les fois qu’ils occuperont le meme rang 
dans la premiere suite et dans la nouvelle, et en formant un 
groupe isole de chaque terme qui n’aura pas change de rang 
dans le passage d’une suite a l’autre.” 
The question of the natural order of ideas and the best mode of 
presentment is really, however, of small importance, for in applica- 
tion a group and a circular substitution are essentially the same. 
The difference is entirely one of stand-point, nomenclature, and 
notation. The permutation 
e, a, b, d } c, g, f, 
I.e., permutations of a,b,c,d, . . . 
