An Elementary Treatise on Elliptic Functionsby
In the short Preface, Cayley relates the work to other approaches to the topic but gives little indication of the contents, letting the Contents page speak for itself:
The present treatise is founded upon Legendre's Traité des Fonctions Elliptiques, and upon Jacobi's Fundamenta Nova, and Memoirs by him in Crelle's Journal: comparatively very little use is made of the investigations of Abel or of those of later authors. I show how the transition is made from Legendre's Elliptic Integrals of the three kinds to Jacobi's Amplitude, which is the argument of the Elliptic Functions (the sine, cosine, and delta of the amplitude, or as with Gudermann I write them, sn, cn, dn), and also of Jacobi's functions Z, P, which replace the integrals of the second and third kinds, and of the functions Q, H, which he was thence led to. It may be remarked as regards the Fundamenta Nova, that in the first part Jacobi (so to speak) hurries on to the problem of transformation without any sufficient development of the theory of the elliptic functions themselves; and that in the concluding part, starting with the developments furnished by the transformation formulae, he connects with these, introducing them as the occasion arises, his new functions Z, P, Q, H: there are thus various points which require to be more fully discussed. Not included in the Fundamenta Nova we have the important theory of the partial differential equation satisfied by the functions Q, H, and, deduced therefrom, the partial differential equations satisfied by the numerators and denominator in the theories of the multiplication and transformation of the elliptic functions: these I regard as essential parts of Jacobi's theory, and they are here considered accordingly. For further explanation of the range and plan of the present treatise the table of contents, and the first chapter entitled " General Outline," may be consulted. I am greatly indebted to Mr J W L Glaisher of Trinity College for his kind assistance in the revision of the proof-sheets, and for many valuable suggestions.