BIOGRAPHICAL NOTICE OF ARTHUR CAYLEY.
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ndicate parts of the
at least something
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e theory of periodic
at his only book was
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ne classe très-étendue
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ted Cayley’s attention
iis “ Mémoire sur les
oubly-infinite products
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between positive and
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is such as
3 an exponential factor
relation || between the
tance at the time of
of the doubly-infinite
ili complesse, 1868, and in
e other references are given.
Bjerknes, Niels Henrik Abel,
m and n for their Cartesian
', and may be considered to
y the relation between the
l curve.
memoir “ Zur Theorie der
); also in his book Abhand-
a function the value of which is independent of any particular form of relation between
the infinities of m and of n. Owing to the latter simplification, Cayley’s results are,
as he himself remarked*, partly superseded by those of Weierstrass.
Cayley had great admiration for the works of both Abel and Jacobi; he had
begun to read the latter’s Fundamenta Nova immediately after his degree. The
prominent position occupied in that work by the theory of transformation naturally
attracted his interest; and, even as early as 1844 and 1846, he wrote short memoirs
upon the subject, obtaining in one of them a function, due to Abel and now known
as the octahedral function. Further memoirs of a similar tenor appeared occasionally;
they deal chiefly with transformation as concerned with the known differential relation
of the form
{(1 — x 2 ) (1 — k 2 x 2 ))dx = M{(1 — y 2 ) (1 — \ 2 y 2 )}~% dy.
The contributions made to the transformation theory by Sohnke, Joubert, and Hermite,.
as well as Jacobi’s original investigations, all depend upon the use of transcendental
A
functions of the quantity q(=e n K ): yet the results are such that they ought to be
deducible by ordinary algebraical processes. It was Cayley’s wish to deal with this
theory by pure algebra; two simple cases had already thus been discussed by Jacobi,
but the extension to the less simple cases proved difficult. Cayley’s “Memoir on the
transformation of elliptic functions*!*,” carries on the algebraical theory and places it in
a clearer light than before. But though he made a distinct advance in dealing with
particular cases, he still found it necessary to use the ^-transcendents for making any
definite advance in the general case. And the same compulsion occurs in the chapters
of his Treatise on Elliptic Functions, where transformation is discussed at considerable
length.
He resumed his investigations in 1886, still dealing with the algebraical method,
but applying it to a simplified form of elliptic integral due to Brioschi. Though the
problem is not solved £ completely for the general case, he has devised a method which
is effective at least in part; it easily leads to new results connected with the modular
equations in the known simpler cases previously solved.
The theta-functions are the subject of several of his papers. He began § with a
direct establishment of Jacobi’s relation
V& snu = H (u) -r- © (u),
obtained in the Fundamenta Nova by a long and cumbrous process; and he proceeded
to the construction of the linear differential equations satisfied by the theta-functions.
Except, however, in so far as they arise in the transformation theory, they do not
appear to have occupied him until about 1877. In that year and in the succeeding * * * §
* c. M. P. vol. i. p. 586.
f C. M. P. vol. IX. No. 577; Phil. Trans. 1874, pp. 397—456.
J The memoirs of this period belonging to the transformation of elliptic functions were published in the
American Journal of Mathematics, vol. ix. (1887), pp. 193—224; vol. x. (1888), pp. 71—93.
§ “On the Theory of Elliptic Functions,” C. M. P. vol. i. No. 45; Camb. and Dubl. Math. Jour. vol. n.
(1847), pp. 256—266.
