The function which is discussed in the greater part of this paper was introduced by Professor E. T. Whittaker in vol. xxxv (1915), pp. 181–194 of these Proceedings. A one-valued function, analytic in the finite part of the plane save possibly for poles, is taken and a table made of the values of f(x) for x = a, a + w, a — w, a + 2w, a — 2w, … The functioncalled the Cardinal Function, is then shown to be the simplest function satisfying the table of values so obtained, in that it is a function which, when analysed into its periodic constituents, has no rapid oscillations.