The Collected Mathematical Papers of Arthur Cayley, ScD, FRS

Abstract

THIS instalment of the papers illustrates in a remarkable way Cayley's power of commenting upon and developing the work of his predecessors. The various memoirs on single and double theta-functions are, of course, based upon the results of Rosenhain, Göpel, and Kummer; and it is instructive to see how Cayley's instinct for symmetry and logical consistency has enabled him to present the theory in a compact and intelligible form. In the case of the single theta-functions, defined by their expansions in series, we have equations such as and from these it appears that any three of the squared functions θ2gh(u) are connected by a linear relation. Hence we may take the squared functions to be proportional to A(a-x), B(b-x), (c-x), D(d-x) with x a variable, and the other quantities constant. Finally it is shown that x and u are connected by a differential equation of the form The Collected Mathematical Papers of Arthur Cayley, Sc.D., F.R.S. Vols. x., xi. Pp. xiv + 616; xvi + 644. (Cambridge: at the University Press, 1896.)