^{page 300 note *} Cf. V. Volterra, Ann. Sc. Normale di Pisa (1883). Levi Civita, T., Turin Memoirs, t. 49 (1899), p. 105Google Scholar, classifies the binary potentials.

^{page 300 note †} Cf. Störmer, C., Comptes Rendu, Paris, t. 146 (1908), pp. 462Google Scholar, 623.

^{page 300 note ‡} These solutions have been discussed by Jacobi, Forsyth, Levi Civita, and the author. See the author's Electrical and Optical Wave Motion, pp. 114, 136, 153. Also Annals of Mathematics, vol. xiv (1912), p. 51. Solutions of type V = ZF(X, Y), where F satisfies a partial differential equation, have been discussed by Amaldi, , Bend. Palermo, t. 16 (1902), pp. 1–45.CrossRefGoogle Scholar See also Häntzschel, Reduction der Potentialgleichung.

^{page 300 note §} These solutions are of interest in the problem of transforming a special type of electromagnetic field into an electrostatic field. One solution of this problem is derived by setting up a Lorentzian transformation between X, Y, Z, T and x, y, z, t. As far as I know, a solution of this problem is not necessarily associated with a set of functions of type X, Y, Z; for instance, I have not yet succeeded in determining the special type of electromagnetic field associated with the functions X = x cos wt — y sin wt, Y = x sin wt + y cos wt, Z = z — vt, where v and w are constants.

^{page 300 note ‖} Messenger of Mathematics (1914), p. 164. Solutions of type V = WF(X, Y, Z), where F satisfies a partial differential equation, have been discussed in a paper by the author, Cambr. Phil. Trans. (1910), vol. xxi, p. 257.

^{page 301 note *} Electrical and Optical Wave Motion, pp. 124–127. The theorem is given here in a slightly different form.

^{page 302 note*} Lie, Sophus, Theorie der Transformationgruppen, Bd. ii, p. 460.Google Scholar

^{page 302 note †} See Whittaker's Analytical Dynamics, p. 8.

^{page 303 note *} Proc. Roy. Soc. London, vol. lvi (1894), p. 46; “Papers printed to commemorate the Incorporation of the University College of Sheffield” (1897), pp. 60–88. The formula which I have used recently (Terrestrial Magnetism, Sept. 1915) for the mean value of a function round a circle can be deduced immediately from a result given in the first of these papers.

^{page 303 note †} Zeitschr. für Math. u. Phys., Bd. xliv (1899), p. 327.

^{page 303 note ‡} Cf. Hobson, E. W., Encyclopædia Britannica, vol. xxv, p. 651.Google Scholar

^{page 303 note §} See the author's Electrical and Optical Wave Motion, p. 112.

^{page 304 note *} Whittaker's Analytical Dynamics, p. 11.

^{page 304 note †} This formula was needed in an investigation on the Diurnal Variation of Terrestrial Magnetism, made during the summer of 1915 at the Department of Terrestrial Magnetism of the Carnegie Institution of Washington.

^{page 304 note ‡} Mémoire sur l'attraction des sphéroides, Paris (1816).

^{page 305 note *} To pass from Leahy's notation to ours we must put 2u = p.

^{page 305 note †} Crelle's Journal, Bd. lvi (1859), p. 156; Werke, Bd. vi, p. 191.

^{page 305 note ‡} See the author's Electrical and Optical Wave Motion, p. 109, Ex. 1.

^{page 305 note §} Bateman, H., Proc. London Math. Soc., ser. 2, vol. iii (1905), p. 123.Google Scholar

^{page 306 note *} Phil. Trans., A, vol. cciv (1904), p. 481.

^{page 306 note †} Monthly Notices of the Royal Astron. Soc., vol. lxii (1902), p. 617; Math. Ann., Bd. lvii (1903), p. 333.

^{page 307 note *} For a definition of this function see a paper by the author, Bulletin of the American Mathematical Society, April (1916).

^{page 308 note *} This means that the equations div_p grare_q are satisfied by each component of M.

^{page 309 note *} Bulletin de la Société mathematique de France, t. 19, p. 68.

^{page 309 note †} This means that each component of L is a right-handed multiple wave-function.

^{page 310 note *} Bulletin of the American Mathematical Society, April (1916).

^{page 312 note *} Math. Ann., Bd. lvii (1903).