The transformation of partial differential equations
Author:
H. Bateman
Journal:
Quart. Appl. Math. 1 (1944), 281-296
MSC:
Primary 36.0X
DOI:
https://doi.org/10.1090/qam/9686
MathSciNet review:
9686
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- J. E. Campbell, Einstein’s Theory of Gravitation as an Hypothesis in Differential Geometry, Proc. London Math. Soc. (2) 20 (1921), no. 1, 1–14. MR 1577354, DOI https://doi.org/10.1112/plms/s2-20.1.1
C. Carathéodory, Bull. Greek Math. Soc. 3, 16–24 (1922); Math. Ann. 86, 272–275 (1922); 85, 78–88 (1922).
G. Monge, Application de l’analyse à la géométrie, Bernard, Paris, 1809.
A. Haar, J. f. Math. 149, 1–18 (1919); see also T. Radó, Hungarian Univ. Acta, Szeged, 2, 147–156 (1925).
A. M. Legendre, Histoire de l’Académie des Sciences 1787, 309.
H. Hencky, Zeit. f. angew. Math. u. Mech. 3, 241–251 (1923).
L. Prandtl, idem., 1, 15–20 (1921); 3, 401–408 (1923).
C. Carathéodory and E. Schmidt, idem. 3, 468–475 (1923).
C. W. Oseen, Arkiv f. Math., Ast., o Fysik 20A, Nos. 24, 25 (1928).
J. R. Wilton, Messenger of Math. (2) 43, 58–63 (1914).
A. V. Bäcklund, Math. Ann. 9, 109–116 (1876).
S. Lie, Math. Ann. 59, 193–313 (1904).
J. Kürschák, Math. Ann. 61, 109–116 (1905).
G. Cerf, Transformations de contact et problème de Pfaff, Mémorial Sci. Math., Fasc. 37, Gauthier-Villars, Paris, 1929, p. 4.
F. Engel, Deutsch Math. Verein, Jahresb. Ergänzungsband 5, 4 (1914); see also G. Darboux, Bull, des Sci. Math. 6, 14–36, 49–68 (1882).
E. Goursat, Comptes Rendus 134, 459–462, 1035–1038 (1902); Ann. Fac. Sci. Toulouse (2) 4, 299–304 (1902); Leçons sur l’intégration des équations aux derivées partielles du second ordre, Hermann, Paris, 1896; Le problème de Bäcklund, Mémorial Sci. Math., Fasc. 6, Gauthier-Villars, Paris, 1925.
F. J. Clairin, Ann. de l’Ecole Normale (3) 19 suppl., 3–63 (1902); Toulouse Ann. (2) 5, 437–458 (1903); Comptes Rendus 153, 39–40 (1911); 154, 1579–1581 (1912); 158, 923–926 (1914); Bull. Math. Soc. France 41, 206–228 (1913); Ann. de l’Ecole Normale (3) 30, 173–192 (1913). See also S. Büschgens, Moscow Math. Soc. 26, 24–36 (1906); S. Zervos (Drakos), Comptes Rendus 171, 781–783 (1920); Bull. Greek Math. Soc. 6, 33–34 (1925).
A. R. Forsyth, Theory of differential equations, Part 4, vol. 6, Cambridge University Press, Cambridge, 1906, p. 432–441.
H. Bateman, Proc. Nat. Acad. Sci. 24, 246–251 (1938).
A. Haar, Math. Ann. 100, 481–502 (1928).
E. Gau, Comptes Rendus 156, 116–118 (1913); 158, 676–679 (1914); 176, 278–280 (1923).
G. Darboux, Théorie générale des surfaces, vol. 2, Gauthier-Villars, Paris, 1889.
L. Bianchi, Vorlesungen über Differentialgeometrie (trans, by Lecat), Teubner, Leipzig, 1910.
L. P. Eisenhart, Transformation of surfaces, Princeton University Press, Princeton, 1923.
E. Cotton, Comptes Rendus 123, 936–938 (1896); Ann. de l’Ecole Normale (3), 17, 211–244 (1900).
P. Rivereau, Bull. Soc. Math. France 29, 7–15 (1901).
P. S. Laplace, Mémoires de l’Acad. Royale des Sciences 1773, 341; Oeuvres complètes de Laplace, t. 9, Gauthier-Villars, Paris, 1878–1912, p. 5–68; A. R. Forsyth, loc. cit. [18], Ch. XIII; E. Goursat, loc. cit. [16], No. 3, Ch. 5.
M. Chini, Torino Atti 31, 568–578 (1896).
P. Burgatti, Annali di Mat. (2) 23, 225–267 (1895).
A. E. H. Love and F. B. Pidduck, Trans. Roy. Soc. London A 222, 167–226 (1922).
C. Platrier, Comptes Rendus 199, 770–772 (1934); Mémorial de l’Artillerie Française 3, 431–477 (1934).
K. Bechert, Annalen d. Physik (5) 37, 89–123 (1940); 38, 1–25 (1940); 39, 169–202, 357–372 (1941).
P. Molenbroek, Arch. Math. Phys. (2) 9, 157–195 (1890).
S. Tschaplygin, Gas Jets, Scientific Memoirs, Moscow Univ., 1902, 1–121.
E. Beltrami, Giorn. di Mat. 2, 267–288, 297–306, 355–375, 331–339 (1864); 3, 15–22, 33–41, 82–91, 225–240 (1865); Opere matematiche, vol. 1, U. Hoepli, Milan, 1902, p. 107–198.
G. Ricci and T. Levi-Civita, Math. Ann. 54, 125–201 (1901).
T. Levi-Civita, Ven. Ist. Atti. 72, 1331–1357 (1913).
G. Lamé, Leçons sur les coordonnées curvilignes et leurs diverses applications, Mallet Bacelier, Paris, 1859.
M. Bôcher, Über die Reihenentwicklungen der Potentialtheorie, Teubner, Leipzig, 1894. See also A. Wangerin, Akad. d. Wiss. Berlin, Monatsb. 1878, 152–166.
J. Larmor, Trans. Camb. Phil. Soc. 14, 121–137 (1886); Mathematical and physical papers, vol. 1, Cambridge University Press, Cambridge, 1929, p. 113–132.
V. Volterra, Rend. Lincei (4) 5, 599–611, 630–640 (1889); (4) 6, 43–54 (1890); G. Darboux, loc. cit. [22], vol. 3; M. Fujiwara, Tôhoku Math. J. 1, 8–18 (1911); J. Kürschák, Math. Ann. 56, 155–164 (1903); L. Koschmieder, Math. Z. 24, 181–190 (1926); 25, 74–86 (1926).
G. Stächel, Comptes Rendus 116, 485–487 (1893).
L. P. Eisenhart, Annals of Math. (2) 35, 284–305 (1934).
H. P. Robertson, Math. Ann. 98, 749–752 (1928).
V. Volterra, Tesi di abilitazione, Annali Scuola Normale di Pisa (1883), T. Levi-Civita, Torino Mem. 49, 105–152 (1899); H. Bateman, Trans. Camb. Phil. Soc. 21, 257–280 (1910).
J. E. Campbell, Proc. London Math. Soc. (2) 10, 406–422 (1910).
C. Carathéodory, Bull. Greek Math. Soc. 3, 16–24 (1922); Math. Ann. 86, 272–275 (1922); 85, 78–88 (1922).
G. Monge, Application de l’analyse à la géométrie, Bernard, Paris, 1809.
A. Haar, J. f. Math. 149, 1–18 (1919); see also T. Radó, Hungarian Univ. Acta, Szeged, 2, 147–156 (1925).
A. M. Legendre, Histoire de l’Académie des Sciences 1787, 309.
H. Hencky, Zeit. f. angew. Math. u. Mech. 3, 241–251 (1923).
L. Prandtl, idem., 1, 15–20 (1921); 3, 401–408 (1923).
C. Carathéodory and E. Schmidt, idem. 3, 468–475 (1923).
C. W. Oseen, Arkiv f. Math., Ast., o Fysik 20A, Nos. 24, 25 (1928).
J. R. Wilton, Messenger of Math. (2) 43, 58–63 (1914).
A. V. Bäcklund, Math. Ann. 9, 109–116 (1876).
S. Lie, Math. Ann. 59, 193–313 (1904).
J. Kürschák, Math. Ann. 61, 109–116 (1905).
G. Cerf, Transformations de contact et problème de Pfaff, Mémorial Sci. Math., Fasc. 37, Gauthier-Villars, Paris, 1929, p. 4.
F. Engel, Deutsch Math. Verein, Jahresb. Ergänzungsband 5, 4 (1914); see also G. Darboux, Bull, des Sci. Math. 6, 14–36, 49–68 (1882).
E. Goursat, Comptes Rendus 134, 459–462, 1035–1038 (1902); Ann. Fac. Sci. Toulouse (2) 4, 299–304 (1902); Leçons sur l’intégration des équations aux derivées partielles du second ordre, Hermann, Paris, 1896; Le problème de Bäcklund, Mémorial Sci. Math., Fasc. 6, Gauthier-Villars, Paris, 1925.
F. J. Clairin, Ann. de l’Ecole Normale (3) 19 suppl., 3–63 (1902); Toulouse Ann. (2) 5, 437–458 (1903); Comptes Rendus 153, 39–40 (1911); 154, 1579–1581 (1912); 158, 923–926 (1914); Bull. Math. Soc. France 41, 206–228 (1913); Ann. de l’Ecole Normale (3) 30, 173–192 (1913). See also S. Büschgens, Moscow Math. Soc. 26, 24–36 (1906); S. Zervos (Drakos), Comptes Rendus 171, 781–783 (1920); Bull. Greek Math. Soc. 6, 33–34 (1925).
A. R. Forsyth, Theory of differential equations, Part 4, vol. 6, Cambridge University Press, Cambridge, 1906, p. 432–441.
H. Bateman, Proc. Nat. Acad. Sci. 24, 246–251 (1938).
A. Haar, Math. Ann. 100, 481–502 (1928).
E. Gau, Comptes Rendus 156, 116–118 (1913); 158, 676–679 (1914); 176, 278–280 (1923).
G. Darboux, Théorie générale des surfaces, vol. 2, Gauthier-Villars, Paris, 1889.
L. Bianchi, Vorlesungen über Differentialgeometrie (trans, by Lecat), Teubner, Leipzig, 1910.
L. P. Eisenhart, Transformation of surfaces, Princeton University Press, Princeton, 1923.
E. Cotton, Comptes Rendus 123, 936–938 (1896); Ann. de l’Ecole Normale (3), 17, 211–244 (1900).
P. Rivereau, Bull. Soc. Math. France 29, 7–15 (1901).
P. S. Laplace, Mémoires de l’Acad. Royale des Sciences 1773, 341; Oeuvres complètes de Laplace, t. 9, Gauthier-Villars, Paris, 1878–1912, p. 5–68; A. R. Forsyth, loc. cit. [18], Ch. XIII; E. Goursat, loc. cit. [16], No. 3, Ch. 5.
M. Chini, Torino Atti 31, 568–578 (1896).
P. Burgatti, Annali di Mat. (2) 23, 225–267 (1895).
A. E. H. Love and F. B. Pidduck, Trans. Roy. Soc. London A 222, 167–226 (1922).
C. Platrier, Comptes Rendus 199, 770–772 (1934); Mémorial de l’Artillerie Française 3, 431–477 (1934).
K. Bechert, Annalen d. Physik (5) 37, 89–123 (1940); 38, 1–25 (1940); 39, 169–202, 357–372 (1941).
P. Molenbroek, Arch. Math. Phys. (2) 9, 157–195 (1890).
S. Tschaplygin, Gas Jets, Scientific Memoirs, Moscow Univ., 1902, 1–121.
E. Beltrami, Giorn. di Mat. 2, 267–288, 297–306, 355–375, 331–339 (1864); 3, 15–22, 33–41, 82–91, 225–240 (1865); Opere matematiche, vol. 1, U. Hoepli, Milan, 1902, p. 107–198.
G. Ricci and T. Levi-Civita, Math. Ann. 54, 125–201 (1901).
T. Levi-Civita, Ven. Ist. Atti. 72, 1331–1357 (1913).
G. Lamé, Leçons sur les coordonnées curvilignes et leurs diverses applications, Mallet Bacelier, Paris, 1859.
M. Bôcher, Über die Reihenentwicklungen der Potentialtheorie, Teubner, Leipzig, 1894. See also A. Wangerin, Akad. d. Wiss. Berlin, Monatsb. 1878, 152–166.
J. Larmor, Trans. Camb. Phil. Soc. 14, 121–137 (1886); Mathematical and physical papers, vol. 1, Cambridge University Press, Cambridge, 1929, p. 113–132.
V. Volterra, Rend. Lincei (4) 5, 599–611, 630–640 (1889); (4) 6, 43–54 (1890); G. Darboux, loc. cit. [22], vol. 3; M. Fujiwara, Tôhoku Math. J. 1, 8–18 (1911); J. Kürschák, Math. Ann. 56, 155–164 (1903); L. Koschmieder, Math. Z. 24, 181–190 (1926); 25, 74–86 (1926).
G. Stächel, Comptes Rendus 116, 485–487 (1893).
L. P. Eisenhart, Annals of Math. (2) 35, 284–305 (1934).
H. P. Robertson, Math. Ann. 98, 749–752 (1928).
V. Volterra, Tesi di abilitazione, Annali Scuola Normale di Pisa (1883), T. Levi-Civita, Torino Mem. 49, 105–152 (1899); H. Bateman, Trans. Camb. Phil. Soc. 21, 257–280 (1910).
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