This memoir is devoted to the study of certain new functions, which may be regarded as limiting cases of the “hypergeometric functions of two variables” discovered by Appell.

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§1. Summary. In a very remarkable work on the operational Calculus, Dr Balth. van der Pol 1 has introduced a new function, playing with respect to Bessel function of order zero the same part as the cosine-or sine-integral with respect to the ordinary cosine or sine. He showed that this function-which he called Besselintegral junction-can be used to express the differential coefficient of any Bessel function with respect to its index. But he did not investigate the further properties of his new function. I propose to give here some of them, which appear to be interesting, and to introduce and study the functions connected, in the same way, with Bessel functions of any order. § 2. Definition of the function of order zero.

The polynomials which satisfy linear differential equations of the second order and of the hypergeometric type have been the object of extensive work, and very few properties of them remain now hidden; the student who seeks in that direction a su' jt-ct for research is compelled to look, not after these functions themselves but after generalisations of them. Among these may be set in first place the polynomials connected with a dift'erential equation of the third order and of the extended hypergeometric type, of which a general theory lias been given by Goursat. The number of such polynomials of which properties have been studied in particular is rather small; in fact, Appell's polynomials and Pincherle's polynomials, arising from the expansions are, so far as I know, the only well-known ones. To show what can be done in these ways, I shall briefly give the definition and principal properties of some polynomials analogous to Pineheile's and of some allied functions. I. Starting from the expressionwhere v is an arbitrary quantity I expand it in ascending powers of t, and call P* n (x) the coefficient of the nth power of t; so P" n is obviously a polynomial of degree n in respect of x; and, bearing in mind the definition of Pincherle's polynomial 1',, given above, it may be seen that P* u plays with 1',, the same part as Gegenbauer's polynomial C" H plays with the ordinary Legendre's polynomial X,,.

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