On Multiple Integration by Parts and the Second Theorem of the Mean

The account of the Formula for Integration by Parts and the Second Theorem of the Mean in Hobson's Treatise is concerned only with functions of a single variable, having Riemann integrals, and a paper by him in these Proceedings, 1909, Ser. 2, Vol. 7, p. 14, confines itself to extensions to Lebesgue integrals, again for the case of a single variable.The Theorem of the Mean for two variables for the integral fg dx dy had been already given in 1903 by M. Krause (Leipziger Bericht, math.Phys.Klasse, pp.239-2G3, " Mittelwertsatze in Gebiete d.Doppelsummon u.Doppelintegrale").Krause supposes/ bounded and integrable (in Riemann's sense, and g a function whose simple and double increments are of constant sign.He also, as he says, for simplicity of exposition, supposes g to be continuous.In 1905, in a paper which I cited on a previous occasion (see "Multiple Fourier Series," Proc.London Math.Soc, Ser. 2, Vol.11, p. 138, 1912), G. H. Hardy considers the same integral for the case in which g is any function of bounded variation, but / is subject to the same restrictions as in the work of Krause.A paper by Lebesgue which appeared in August, 1910, in the Annales de I'Ecole Normale, " Sur l'integration des fonctions discontinues," contains a form of the Second Theorem of the Mean for functions of any number of variables, which I have had occasion to use in a previous communication to the Society.Hardy gives no reference to Krause, and Lebesgue mentions neither Krause nor Hardy.The reference in my paper on "Multiple Fourier Series" is to an earlier paper by Krause, and my acquaintance with the one here referred to is an indirect consequence of a reminder by a kindly referee of those earlier references.In the work of none of these authors, and indeed, as far as I know, of none, since Weierstrass, is the Formula for Integration by Parts utilised in obtaining the Theorem of the Mean.As is well known, Weierstrass confined himself to a single variable, and to a statement of the Theorem of the Mean involving considerable restrictions on the nature of the functions.Cf.Hobson, loc.cit., p. 363.