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Elliptic Integrals

Nature volume 100, page 324 (27 December 1917) | Download Citation

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Abstract

INSPIRED by Sir G. Greenhill, to whom he makes due acknowledgment, Prof. Hancock has compiled a very useful monograph, compact, well arranged, and apparently accurate. Chap. i. is on elliptic integrals, properly so called, and their reduction to Legendre's normal forms; it is illustrated by appropriate graphs. Chap. ii. is on the sn, en, dn functions, and gives the period-pavement for each. Chap. iii. gives a well-arranged list of integrals involving elliptic functions. Chap. iv. is'on computation, and follows Jacobi and Cayley in the main. It begins with Jacobi's two-circle proof of the addition theorem, goes on to the Landen transformation, and then gives worked-put examples, using the descending scale of moduli (k, kv kz, …) as Jacobi does. The algorithm of the arithmetic geometric mean is explained and applied, and there is a particularly neat discussion (p. 79) of integrals of the second kind. There are three tables, all to five places: (i) Complete integrals K, E with fe = sin 6°, and i° step for 9°; (ii) elliptic integrals F(k,) with k as above, step 5° for 6° and i° for 4>°; (iii) elliptic integrals E(k,) with k, as for (ii). All these tables were reproduced from Levy's “Theorie des fonctidns elliptiques “; they are well printed and properly spaced.

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https://doi.org/10.1038/100324a0

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