xl 
BIOGRAPHICAL NOTICE OF ARTHUR CAYLEY. 
is due in part to the extensions he achieved. It is difficult to indicate parts of the 
general theory of surfaces and of twisted curves that do not owe at least something 
and frequently much to his labours; a mere reference to the index of a book like 
Salmon’s Solid Geometry will show how vast has been his influence. 
One group of subjects interested him throughout his life, the theory of periodic 
functions, in particular, of elliptic functions: it was to the latter that his only book was 
devoted. But in a subject, the main lines of which were established so definitely before 
he began to write*, it is impossible, without entering into great detail, to mark out the 
contributions that are directly due to him. When a theory is in such a stage as was 
that of elliptic functions about 1842, the work of one writer sometimes helps to fill the 
gaps left by that of another, sometimes develops another writer’s results from a different 
point of view; the composite theory depends, in part, upon the coordination of comple 
mentary results. 
Abel’s famous paper*f*, “ Mémoire sur une propriété générale d’une classe très-étendue 
de fonctions transcendantes,” presented to the French Academy of Sciences in 1826, and 
unfortunately delayed in publication* for nearly fifteen years, attracted Cayley’s attention 
quite early in his scientific career. In 1845 Cayley published his “Mémoire sur les 
fonctions doublement périodiques,”! in which he considered Abel’s doubly-infinite products 
of the form 
u (x) = ¿dill ("l + — 
where w = (m, n) — mil + nT, the ratio il : T is not real, and the product is taken for 
all positive and all negative integer values of m and of n between positive and 
negative infinity, except simultaneous zero values. He showed that such products can 
be used to obtain Jacobi’s elliptic functions by constructing fractions such as 
u (x + -lil) -t-u(x); 
and he also showed that the actual value of any product involves an exponential factor 
e^ Bx \ where the value of the constant B depends upon the relation || between the 
infinities of m and of n. The results were of definite importance at the time of 
their discovery, and they still hold their place. But the form of the doubly-infinite 
product has been modified IF by Weierstrass, who takes 
* The history will be found in Casorati, Teorica delle funzioni di variàbili complesse, 1868, and in 
Enneper, Elliptische Functionen, Theorie und Geschichte, second edition, 1890, where other references are given. 
t Œuvres complètes d'Abel (Christiania, 1881), vol. i. pp. 145—211. 
Î The circumstances are recited in § 9 of the appendix to the volume, by Bjerknes, Niels Henrik Abel, 
Tableau de sa vie et de son action scientifique (Gauthier-Yillars, Paris, 1885). 
§ C. M. P. voi. i. No. 25 ; Liouville, voi. x. (1845), pp. 385—420. 
Il This is sometimes expressed differently, as follows. Points are taken having m and n for their Cartesian 
co-ordinates ; those which occur for infinite values of m and of n lie at infinity, and may be considered to 
lie upon a curve altogether at infinity, the shape of which is determined by the relation between the 
infinities of m and of n. 
The value of the constant B is said to depend upon the shape of this bounding curve. 
IT Weierstrass’s investigations on infinite products are contained in his memoir “ Zur Theorie der 
eindeutigen analytischen Functionen ” (Abh. d. K. Akad. d. Wiss. zu Berlin, 1876) ; also in his book Abhand- 
lungen aus der Functionenlehre, 1886.