429 
1888 - 89 .] Dr T. Muir on the Theory of Determinants. 
( e ) The permutations which arise by compounding a set of 
permutations in every possible order belong all to the 
same class. (in. 31) 
(/) The interchange of two indices is equivalent to the perform- 
ance of a negative permutation. 
(g) The interchange of two indices causes all the positive per- 
mutations to become negative, and all the negative to 
become positive. 
Definition . — Two permutations may be called reciprocal which 
being performed in succession do not alter the order exist- 
ing before the operations. (xxiv. 2) 
(7i) Eeciprocal permutations belong to the same class. 
In the original, it must be borne in mind, these are not separated 
and numbered, but appear merely as consecutive sentences in a 
paragraph. The words “ classem negativam” of the definition 
above given are followed in the same line by 
“ Binis propositis permutationibus quibuscunque, certa ex- 
stabit permutatio, qua post alteram adhibita altera prodit. 
Pertinebunt duse permutationes propositse ad classem eundem 
aut ad classes oppositas, prout permutatio, qua altera ex 
altera obtinetur, ad classem positivam aut negativam pertinet,” 
&c. 
— that is to say, by the propositions which have been paraphrased 
into ( a), ( b ), &c. 
The most essential point to be considered in connection with them 
is the probable meaning of the expression “ permutationem ad- 
hibere,” or the free English translation of it, “ to perform a 
permutation.” An example will make it clear. To perform the 
permutation 35412 would seem to be the operation of removing the 
3rd member of a series of five things to the first place, the 5th 
member to the second place, the 4th member to the third place, 
and so on. With this explanation the proposition (a) is self- 
evident, an example of it being (if we may improvise a symbolism) 
(354 12)(41 352) = (32541), 
where 35412 is the operating permutation. Cauchy’s usage, it may 
