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On chains of two-two relations and the theory of elliptic functions

Published online by Cambridge University Press:  24 October 2008

Extract

1. The relation

in which we suppose each of a, b, c, a2c2 to be different from zero, leads, from a value θ, to two values of φ, which we may denote by θ1 and θ−1. Each of these, put in place of θ, leads, beside the value θ of φ, to another value of φ, say, respectively, θ2 and θ−2. If θ2 or θ−2 be put for θ, the same relation leads to two values of φ, say, θ1 and θ3 or θ−1 and θ−3, respectively. And so on. It may happen that θn is the same as θ, in which case also θ−n is the same as θ; this we may express by saying that the relation is closed, or that there is closure, after n links. It is the object of the present note to express in reduced form, in terms of a, b, c, the condition that this may be so. Evidently, if n = pq, the condition of closure after n links is satisfied when the condition for closure after p links (or also after q links) is satisfied. But there is a condition for closure after n links which is not satisfied for any less number of links; this is the condition which we call the reduced or proper condition for closure after n links, and it is this which we seek to express.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1926

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References

* Cf. Chaundy, , Proc. Lond. Math. Soc. XXII. 1924, p. 110.Google Scholar

Titchmarsh, E. C., Messenger of Math. LII. 1922, p. 42.Google Scholar

* Fonct. Ellipi. II. 1888, p. 377.Google Scholar

* Papers, II. pp. 87, 138.Google Scholar