By L. Auslander, R. Tolimieri
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Extra resources for Abelian Harmonic Analysis, Theta Functions and Functional Analysis on a Nilmanifold
D C 1 ; x x 1 1 d C C1 c bC D x; 1 a x that is to say, xD 1 1 d C C1 c bC 1 . a x This therefore is always the equation of the second degree which gives the two roots in question. But, continually putting for [in place of] x in its second member this same second member, which is, in fact its value, it gives xD 1 1 d C C1 1 c b C C1 1 a d C C1 1 c b C C ::::: . a This is therefore the other value of x given by the equation, a value which, as can be seen, is equal to 1 divided by the first [written in the reverse order].
Période is naturally translated as ‘period’. Note, however, that in the context ‘substitution dont la période sera de p termes’ it means what is now called order—the substitution will have order p. Permutation is naturally translated as ‘permutation’ but with the meaning of an arrangement (static). Substitution is naturally translated as ‘substitution’, meaning an act of rearranging (dynamic); unfortunately, nowadays we use the word ‘permutation’. Substitution circulaire is naturally translated as ‘circular substitution’.
3 The words permutation and substitution The words permutation and substitution are of course translated as ‘permutation’ and ‘substitution’ respectively. Straightforward and natural though that is, it involves pitfalls for the unwary modern reader. The word permutation is ambiguous in French, as it is in English. In English school syllabuses the word ‘permutation’ in the phrase ‘permutations and combinations’ refers to an arrangement of symbols. In undergraduate mathematics it acquires a second (and perhaps more usual) meaning as a bijection of a set to itself.