A BRIEF ACCOUNT
OF THE
HISTORICAL DEVELOPMENT OF
PSEUDOSPHERICAL SURFACES
FROM 1827 TO 1887.
Submitted in partial fulfilment of the bequirements for the degree of
DocTOB OF Philosophy, in the Faculty of Pure Science,
Columbia University.
By
EMILY CODDINGTON.
Press or
The Ne« EflA PaiNTiHC Comfaix
lancasieb, Pa.
1905
*•• • •«•
Sciences i i n )
Library '- ^■f'
A BRIEF ACCOUNT OF THE HISTORICAL DEVELOPMENT OF
PSEUDOSPHERICAL SURFACES FROM 1827 TO 1887
THE APPLICATION OF ONE PSEUDOSPHERICAL SURFACE UPON
ANOTHER AND THE GEOMETRY OF THESE SURFACES.
1. The definition of pseudosplierical surfaces.
2. The definition of total curvature according to Gauss.
3. The application of surfaces with constant curvature upon one another.
4. The pseudospherical surfaces of rotation and the helicoidal surface.
5. Enneper's surfaces.
6. Asymptotic lines on a pseudospherical surface.
7. The identification of the non-Euclidean geometry with pseudospherical geometry.
8. The projections of a pseudospherical surface upon a plane analogous to the central
and stereographic projection of a sphere on a plane.
9. The identification of pseudospherical geometry with the metrical geometry of
Cayley.
II.
THE SURFACE OF CENTERS AND THE TRANSFORBIATION OF ONE
PSEUDOSPHERICAL SURFACE INTO ANOTHER.
1. The theorem of transformation.
2. Kuminer's theory of congruences.
3. Weingarten's two theorems on surfaces whose radii of curvature are functionally
related.
4. Ribaucour's cyclic system of surfaces.
5. Bianchi's complementary transformation.
6. Geodesic lines on the surface of centers.
7. Lie's transformation.
8. Backlund's transformation.
9. Darboux's equations for Bianchi's and Backlund's transformation.
10. The triply orthogonal system of surfaces.
BOOKS OP REFERENCE.
1. Dupin, C, Developpements de geometric. Paris, 1821.
2. Gauss, J., Disquisitiones generales circa surperficies curvas. 1827. Translated
into English by Messrs. Morehead and Hiltebeitel.
3. Minding, E. F. A., Bemerkung iiber die Abwickelung krummer Linien von
Fliichen. J. fiir Math., VI., 1830.
4. Lobatschewsky, Kasaner Boton, 1829. Neue Anfangsgrunde der Geometric nebst
eiuer voUstandigen Theorie der Parallelen. Kasan, 1836-1836. Translated
into French by J. Hoiiel.
1
2 E. M. CODDINGTON.
5. Minding, E. F. A., Ueber die Biegung gewisser Flacheu. J. fiir Math., XVIII.,
1838.
6. Wie sich entscheiden liisst, ob zwei gegebeue krumme Flachen auf ein-
auder abwiekelbar sind oder nicht ; nebst Bemerkuugen iiber die Flachen
von unverauderlichem Kriimmungsmasse. J. fvir Math., XIX., 1839.
7. Zur Theorie der kiirzesteu Linieu auf krummen Flacheu. J. fiir Math.,
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8. Serret, J. A., Note sur une equation aux deriv6es partielles. Jouin. de Math.,
XIII., 1848.
9. Bonnet, 0., M6moire sur la theorie generals des surfaces. J. de TEc. Pol. cah.,
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17. Weingarten, J., Ueber eine Klasse auf einander abwickelbarer Flachen. J. fiir
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21. Beltrami, E., lutorno ad alcune propriety, delle superficie di rivoluzione. Annali
di Mat., VI., 1864.
22. Dini, U., Sulla teoria della superficie. Batt. G., III., 1865.
23. Sulla superficie gobbe che possono, etc. Batt. G., III., 1865.
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25. Des surfaces a courbure costante. C. R., LX., 1865.
26. Beltrami, E., Ricerche di analisi applicata alia geometria. Batt. G., III., 1865.
27. Dimostrazione di due formole del signor Bonnet. Batt. G., IV., 1866.
28. Risoluzione del problema ; Riportare i punti di una superficie sopra un piano
in modo, che le linee geodetiche vengano rappresentate da linee rette.
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29. Dini, U., Sopra alcuni punti della teoria delle superficie applicabili. Annali di
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30. Helmholtz, Ueber die thatsachlichen Grundlagen der Geometrie. 1866.
HISTORICAL DEVELOPMENT OF PSEUDOSPHERICAL SURFACES. 3
31. Bonnet, 0., Memoire sur la theoiie des surfaces applicables suv line surface donn6e.
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32. Beltrami, E., Saggio d'iuterpretazione della geometria noneuelidea. Batt. G. ,
VI., 1S6S. Translated into French by J. Houel, Annales de I'^cole
Kormale, VI.
33. Cliristoffel, E. B., Allgemeine Theorie der geodiitischen Dreiecke. Berl. Abh.,
1868.
34. Dini, U., Sopra alcuni puuti della teoria delle superficie. Mem. della societa
italiaua (3), I., 18G8.
35. Enneper, A., Analytisch-geometrische Untersuchungen. Gott. Nachr., 1868.
36. Die cyklischen Flachen. Schlomilch Z., XIV., 1869.
37. Beltrami, E., Teoria fondamentale degli spazii di curvatura costante. Annali
di Mat. (2), II., 1869.
38. Dini, U., Ricerche sopra la teoria delle superficie. Mem. della societa italiana
(3), II., 1869.
39. Sulle superficie, ehe hanno un sistema di linee di curvatura plane. Ann.
d. Univ. Toscana, 1869.
40. Enneper, A., Ueber eine Erweiterung des Begriflfs von Parallel tlachen. Gott.
Nachr., 1870.
41. Ueber asymptotische Linien. Gott. Nachr., 1870.
42. Untersuchungen iiber einige Puukte aus der allgemeinen Theorie der
Flachen. Math. Ann., II., 1870.
43. Ribaucour, A., Sur la deformation des surfaces. C. R., LXX., 1870.
44. Dini, U., Sulle superficie, che hanno un sistema di linee di curvatura sferiche.
Mem. della societa italiana (3), II., 1870.
45. Klein, F., Ueber die sogenannte Nicht-Euklidische Geometrie. Math. Ann., IV.,
1871.
46. Kretschmer, E., Zur Theorie der Flachen mit ebenen Kriimmungslinien. Progr.
Frankfurt a. O., 1871.
47. Dini, U., Sopra alcune formole generali della teoria delle superficie. Annali di
Blat. (2), IV., 1871.
48. Beltrami, E. , Sulla superficie di rotazione, che serve di tipo alle superficie pseu-
dosferiche. Batt. G., X., 1872.
49. Teorema di geometria pseudosferica. Batt. G., X., 1872.
50. Ribaucour, A., Sur les developpees des surfaces. C. R., LXXIV., 1872.
51. Ribaucour, A., Sur les systemes cycllques. C. R., LXXVL, 1873.
52. Enneper, A., Bemerkungen iiber geodatische Linien. Schlomilch. Z., XVIII.,
1873.
53. Escherich, G. v., Die Geometrie auf den Flachen constanter negativer Kriim-
mung. Wien. Ber., LXIX., 1874.
54. Enneper, A., Bemerkung zu den aualytisch-geometrischen Untersuchungen.
Gott. Nachr., 1874.
55. Klein, F., Ueber Nicht-Euklidische Geometrie. Math. Ann., VII., 1874.
56. Simon, P., Ueber Flachen mit constantem Kriimmungsmass. Diss. Halle, 1876.
57. Enneper, A., Ueber einige Flachen mit constantem Kriimmungsmass. Gott.
Nachr., 1876.
58. Bockwoldt, Gr., Ueber die Enneperchen Flachen mit constantem positivem
Kriimmungsmass. Diss. Gottingen, 1876.
4 E. M. CODDINGTON.
59. Hazzadakis, J. N., Ueber einige Eigenschaften der Flacheu mit constantem
Kriimmungsmass. J. fiir Math., LXXXVIII., 1879.
60. Lie, S., Zur Theorie der Flachen constanter Kriimruung. Arch, for Mat. og.
Naturv., TV., 1879.
61. Ueber Flachen, deren Kriimmungsradieu durch eiue Relation verkuiipft
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62. Klassifikatiou der Flachen nach der Transformatiousgruppe ihrer geoda-
tischen Curven. Universitatsprogr. Christiania, 1879.
63. Bianchi, L., Ricerche sulle superficie a curvatura costante e suUe elicoidi.
Aunali di Pisa, II., 1879.
64. Cayley, A., On the correspondence of homographies and rotations. Math. Ann.,
XY., 1879.
65. Lecornu, Sur I'equilibre des surfaces flexibles et inextensibles. J. de I'Ec. Polj'.,
XXIX., 1880.
66. Lenz, Ueber die Enneperschen Flachen constanter negativer Krummung. Diss.
Gottingen, 1879.
67. Lie, S., Sur les surfaces, dont les rayons de courbure ont entre eux une relation.
Darboux Bull. (2), IV., 1880.
68. Zur Theorie der Flachen constanter Kriimmung. Ai-ch. for Mat. og
Naturv., Y., 1880.
69. Bianchi, L., Ueber der Flachen mit constanter negativer Kriimmung. Math.
Ann., XVI., 1880.
70. Backlund, A. V., Zur Theorie der partiellen DifFerentialgleichungen. Math.
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71. Voss, A., Ueber ein Priucip der Abbildung krummer Oberflachen. Math. Ann.,
XIX., 1881.
72. Nebelung, Trigonometrie der Flachen mit constantem Kriimmungsmass. Pr.
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73. Lie, S., Trausformationstheorie der partiellen Differentialgleichuug.
,= _,, = (l±^!+i!)!.
Arch, for Mat. og. Naturv., VI., 1881.
74. Lie, S., Diskussion der Differentialgleichuug s = F{z). Arch, for Mat. og. Naturv.,
VI., 1881.
75. Backlund, A. V., Zur Theorie der Fliichentransformationen. Math. Ann., XIX.,
1881.
76. Haas, A., Vorsuch einer Darstellung der Geschichte des Kriimmungsmasses. 1881.
77. Bianchi, L., Sulle superficie a curvatura costante positiva. Batt. G., XX., 1882.
78. Weingarten, J., Ueber die Verschiebbarkeit geodatischer Dreiecke in krummen
Flachen. Berl. Ber., 1882.
79. Ueber die Eigenschaften des Linienelementes der Flachen von constantem
Kriimmungsmass. J. fiir Math., XCIV. u. XCV., 1883.
80. Darboux, G., Sur les surfaces, dont la courbure totale est constante. C. R.,
XCVII., 1883.
81. Sur I'^quation aux derivees partielles des surfaces a courbure constante.
C. R., XCVII., 1883.
82. Mangoldt, H. v., Klassifikatiou der Flacheu nach der Verschiebbarkeit ihrer
geodatischen Dreiecke. J. fiir Math., XCIV., 1883.
HISTORICAL DEVELOPMENT OF PSEUDOSPHERICAL SURFACES. 5
83. Bianclii, L., Sopra alcune classi di sistema tripli ciclici di superficie ortogonali.
I?;Ut. O., XXI., 1SS3.
84. Lie, S.iUntei-suchungenuberDifferentialgleichungen. ChristianiaForh., XVT^II.,
1883.
85. Kuen, Th., Ueber Flachen von constantem Kriimmungsmass. Munch. Ber., 1884.
86. Baicklund, A. V., Om ytor med koustant uegativ krokuing. Lund. Arsskr., 1884.
87. Cayley, A., On the nou-Euclidiau plane geometry. Loudon R. Soc. Proceed.,
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88. Bianchi, L., Sui sistemi tripli ciclici di superficie ortogonali. Batt. G., XXII.,
1884.
89. Sopra una classe di sistemi tripli di superficie ortogonali. Annali di Mat.
(2), XIII., 1885.
90. Sopra i sistemi tripli ortogonali di Weingarten. Annali di Mat. (2), XIII.,
1885.
91. Sopra i sistemi tripli ortogonali di Weingarten. Rom. Ace. L. Rend. (4),
I., 1885.
92. Volterra, V., Sulla deformazione delle superficie flessibili ed inestendibili. Rom.
Ace. L. Rend. (4), I., 1885.
93. Bianchi, L., Aggiunte alia memoria ; " Sopra i sistemi tripli ortogonali di Wein-
garten." Annali di Mat. (2), XIV., 1886.
94. Sopra i sistemi tripli ortogonali, che contengono uu sistema di superficie
pseudosferische. Rom. Ace. L. Rend. (4), II., 1886.
95. Weingarten, J., Ueber die Deformationen einer biegsamen unausdehnbaren
Flache. J. fur Math., Vol. C, 1886.
96. Dobriner, H., Die Flachen constanter Kriimmung mit einem System spharischer
Kriimmuugslinien. Acta Math., IX., 1886.
97. Lipschitz, R., Ueber die Oberflachen, bei denen die Differenz der Hauptkrum-
mungsradien constant ist. Acta Math., X., 1887.
98. Bianchi, L., Sopra i sistemi doppiamente infiniti di raggi. Rom. Ace. L. Rend.
(4), III., 1887.
99. Sui sistemi doppiamente infiniti di raggi. Annali di Mat. (2), XV., 1887.
100. Oekinghaus, E., Ueber die Pseudosphiire. Hoppe Arch. (2), V., 1887.
101. Lilienthal, R. v., Zur Theorie der Kriimmungsmittelpuuktsflachen. Math. Ann.,
XXX., 1887.
102. Bemerkung uber diejenigen Flachen, bei denen die Diflferenz der Haupt-
kriimmungen constant ist. Acta Math., XL, 1888.
103. Nannei, E., Le superficie ipercicl'che. Napoli Rend. (2), II., 1888.
104. Le superficie ipercicliche. Batt. G., XXVI., 1888.
105. Marcolongo, R., Sulla rappresentazione conforme della pseudofera e sue applica-
zioni. Napoli Rend. (2), II., 1888.
106. Bazzaboni, A., Sopra certe famiglie di superficie di rivoluzione applicabili.
Bologna Ace. Rend., 1888.
107. Pirondini, G., Studio sulle superficie elicoidali. Annali di Mat. (2), XVI., 1888.
108. Weingarten, J., Ueber eine Eigenschaft der Flachen, bei denen der eine Haupt-
kriimmungsradius eine Function des anderen ist. J. fiir Math., GUI., 1888.
109. LiouviUe, R., Sur les lignes geodesiques des surfaces a courbure constante.
American J., X., 1888.
110. Voss, A., L^eber diejeuigen Flachen, auf denen 3 Scharen geodatischer Linien eiu
conjugirtes System bllden. Munch Ber., 1888.
6 E. M. CODDINGTON.
111. Koenigs, G., Sur Ics surfaces, dont le da" peut etre rameue de plusieurs manieres
au type de Liouville. C. R., CIX., 1889.
112. Raflfy, L., Sur im probleme de la th^orie des surfaces. Darboux Bull., XIII., u.
C. R., CVIII., 1889.
113. Chini, M., Sulle superfieie a curvatura media costante. Batt. G., XXVII., 1889.
11-t. Guichard, C, Surfaces rapportees a leurs lignes asjinptotiques. Ann. de \"kc.
Norm., VI., 1889.
11.5. Beina, V., Sugli oricicli delle superfieie pseudosfericlie. Rom. Ace. L. Rend. (4),
V,., 1889.
116. Razziboni, A., Delle superfieie, sulle quale due serie di goedetiche formano un
sistema couiugato. Bologna Mem. (4), IX., 1889.
117. Sulla rappresentazione di una superfieie su di un altra al modo di Gauss.
Batt. G., XX^^I., 1889.
118. Bianchi, L., Ricerche sulle superfieie elicoidali. Batt. G., XXVII., 1889.
119. Guichard, C, Recherches sur les surfaces a courbure totale costante. Ann. de
I'Ec. Norm. (3), \T;I., 1890.
120. - — sur les surfaces, qui possedent un rfeeau de g^od^siques conjugu^es. C.
E., ex., 1890.
121. Padova, E., Sopra un teorema di geometria differenziale. 1st. Lomb. Rend.,
XXIII., 1890.
122. Bianchi, L., Sopra una classe di rappresentazioni equivalenti della sfera sul piano.
Rom. Ace. L. Rend. (4), Y\., 1890.
123. Sopra una nuova classe di superfieie appartenenti a sistemi tripli orto-
gonali. Rom. Ace. L. Rend. (4), VI,., 1890.
124. Sulle superfieie, le cui linee assintotiche in un sistema sono curve a
torsione costante. Rom. Ace. L. Rend. (4), VIj., 1890.
125. Sopra alcune nuove classi di superfieie e di sistemi tripli ortogonali.
Anuali di Mat. (2), XVIII., 1890.
126. Schwarz, H. A., Gesammelte Abbandlungen. II. Bd., S. 363 ff. 1890.
127. Picard, E., Th^orie des equations aux derivees partielles. Journ. de Math. (4),
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128. Ribancour, A., Memoire sur la theorie generale des surfaces eourbes. Journ. de
Math. (4), VII., 1891.
129. Voss, A., Zur theorie der Krummung derFlachen. Math. Ann., XXXIX., 1891.
130. Cosserat, E., Sur les systemes cycliques et la deformation des surfaces. C. R.,
CXIII., 1891.
131. Sur les sj-st^mes conjugues et sur la deformation des surfaces. C. R ,
CXIII., 1891.
132. Bianchi, L., Sui sistemi tripli ortogonali, che eontengono una serie di superfieie
con un sistema di liuee di curvatura plane. Annali di Mat. (2), XIX., 1891.
133. Wangerin, A., Ueber die Abwickelung von Rotationsflachen mit constantem
negativem Kriimmungmass auf einander. Naturf. Ges. Halle, 1891.
134. HoUaender, E., Ueber aiquivalente Abbildung. Diss. Halle u. Progr. Miihlheim
a. d. Ruhr, 1891.
135. Padova, E., Di alcune classi di superfieie suscettibili di deformazioni iufinitesime
speciali. 1st. I^omb. Rend. (2), XXIV., 1891.
136. Backlund, A. V., Anwendung von Satzen iiber partielle DifTereutialgleichungen
auf die Theorie der Orthogonalsysteme. Math. Ann., XL., 1892.
HISTORICAL DEVELOPMENT OF PSEUDOSPHERICAL SURFACES. 7
137. Bianchi, L., Sulla trasfoimazioue di Backluiul per le superficie pseudosferiche.
Rom. Ace. L. Rend. (5), I^., 1892.
138. Sulla trasformazione di Biickluud pei sistemi tripli ortogouali pseudo-
sferici. Rom. Ace. L. Rend. (-5) I,.. 1892.
139. Sulle deformazioni iufinite.sime delle superficie flessibili ed inestendibili.
Rom. Ace. L. Rend. (5), I^., 1892.
140. Koenings, G-., Resume d'un memoire sur les lignes geodesiques. Toulouse
Auiiales, VI., 1892.
141. Cosserat, E., Sur la deformation infinitdsimale et sur les surfaces assocides de
Bianchi. C. R., CXV., 1893.
142. Sur les congruences des droites et sur la th^orie des surfaces. Toulouse
Annales, VII., 1893.
143. Guichard, C, Sur les surfaces, dout les plans principaux sont equi distants d'un
point fixe. C. R., CXVI., 1893.
144. Waelsch, E., Sur les surfaces a Element lineaire de Liouville et les surfaces a
courbure constante. C. R., CXVI., 1893.
14.5. Ueber die Flachen constanter Kriimmung. Wien Ber., CII., 1893.
146. Probst, F., Ueber Flachen mit isogonalen Systemeu von geod tischeu Kreisen.
Diss. Wiirzburg, 1893.
147. BiancM, L., Sulla interpretazione geometrica del teorema di Moutard. Rom.
Ace. L. Rend. (5), III„., 1894.
148. Applicazioni geometriche del metodo delle approssimazioni successive di
Picard. Rom. Ace. L. Rend. (5). III,., 1894.
149. Sulle superficie, i cui piani principali hanno costante il rapporto delle
distanze da un punto fisso. Rom. Ace. L. Rend. (5), IIIj., 1894.
150. Sui sistemi tripli ortogonali di Weingarten. Palermo Rend., VIII., 1894.
151. Soler, E., Sopra una certa deformata della sfera. Palermo Rend., VIII., 1894.
152. Cosserat, E., Sur des congruences rectilignes et sur le problfeme de Ribaucour.
C. R., CXVIII., 1894.
153. Genty, E., Sur les surfaces a, courbure totale constante. S. M. F. Bull., XXII.,
1894.
154. Wangerin, A., Ueber die Abwickelung von Flachen constanten Krummungsmasses
sowie einiger anderer Flachen auf einander. Festschift d. Univ. Halle
1894.
155. Voss, A., Ueber isometrische Flachen. Math. Ann., XLVI., 1895.
156. Filbi, C, Sulle superficie che, da un doppio sistema di traiettorie isogonali sotto
un augelo costante delle linee di curvatura, sono divise in parallelogrammi
infinitesimi equivalentl. Rom. Ace. L. Rend., IVj., 1895.
157. Busse, F., Ueber diejenige punktweise eindeutige Beziehung zweier Flachen-
stiieke auf einander, bei weleher jedergeodatischen Liuie des eiueu eiue Liuie
constanter geodatischer Kriimmung des anderan entspricht. Berl.-Ber.
158. Ueber eine specielle conforme Abbildung der Flachen constanten Kriim-
mungsmasses auf die Ebene. 1896.
159. Darboux, G., Lemons sur la theorie g^uerale des surfaces. Bd. I. -IV., 1887-1896.
160. Bianchi, L., Lezioui di geometria differeuziale. Gott. , 1894.
161. Genty, E., Sur la deformation infinit^simale des surfaces. Toulouse Ann., IX.,
1S96.
162. Goursat, E., Sur les lignes asymptotiques. C. R., CXXII., 1896.
8 E. M. CODDINGTON.
163. Calinon, A., Le theoreme de Gauss sur la courbure. Nouv. Ann., 13, 1895.
164. Weingarten, J., Sur la deformation des surfaces. Jour, de Math. (5), 1896.
165. Bianchi, Nuove ricerche sulla superficie pseudo sferiehe. Annali di Mat., 24,
1896.
166. Sopra una classe die superficie collegate alle superficie pseudo sferiehe.
Rom. Accad. Lin. (5), 5, 1896.
167. Klein, F., Lie's Transformation. Math. Ann., 50, 1897.
168. P. Stackel und Fr. Engel, Gauss, die beideu Bolyai und die Nicht-Euklidische
Geometric. Math. Ann., 49, 1897.
169. Bukreiew, Flachenelement der Flache coustanter Krummung. Kiew Univ.
Nachr, No. 7, 1897.
170. Voss, A., Ueber infinitesimalen Flachen Deformationen. Deutsche Math., Vol. 4,
1896.
171. Zur Theorie der infinitesimalen Biegungdeformationen einer Flache.
Miineh Ber., 27, 1897.
172. Bianchi, L., Sur deux classes de surfaces qui engendrent par un movement heli-
coidal une famille de Lame. Toulous Ann., 11, 1898.
173. Darboux, 0., Legons sur les systemes orthogonaux et les coordin^es curvilignes.
Paris, 1898.
174. Guichard, C, Sur les surfaces S, courbure totale constants. C. R., 126.
175. Sur les systemes orthogonaux et les systemes cycliques. Ann. de I'Ec.
Norn. (3), 15, 1898.
176. Carda, K., Zur Geometrie auf Flachen coustanter Kriimmung. Wein Ber., 107,
1898.
177. Metzler, G. F., Surfaces of rotation with constant measure of curvature. Am. J.
of Math., 20, 1898.
178. Tzitzeia, Sur les surfaces a courbure totale constante. C. R., 128, 1898.
179. Darboux, G., Sur la transformation de M. Lie et les surfaces envelopp6es de
spheres. Darboux Bull. (2), 1897.
180. Kommerell,V.,Bemerkungzurdenasymptotenlinien. Boklen Math. (2), 2, 1900.
181. Waelsch, E., Ueber Flachen mit spharischen oder ebenen Kriimmungslinien.
Hoheschule Briinn, 133, 1899.
182. Hilbert, D., Ueber Flachen von constanter Gaussicher Krummung. Trans. Am.
Math. Soc. (2), 1, 1901.
183. Zuhlke, P., Ueber die geodatischen liuieu und Dreiecke auf den Flachen Con-
stanten Kriimmungsmasses und ihre Beziehung zur die sogenannten nicht-
Euclidischen geometrie. Charlottenburg, 1902.
Note. The above list of books is based upon the one given by Busse at the end of
his essay entitled " Ueher eine specielle conforme Abbildung der Flachen constanten
Kriimmungsmasses auf die £6ene."'^'
In the following pages the notation of the various authors quoted has been trans-
lated into the notation used by Bianchi in his book entitled, Lezioni di geometria differ-
enziale,^^ since the difference in the notation used by the different writers is not of
special interest.
A list of some of Bianchi's formulae is added at the end of this paper.
HISTORICAL DEVELOPMENT OF PSEUDOSPHERICAL SURFACES.
THE APPLICATION OF ONE PSEUDOSPHERICAL SURFACE
UPON ANOTHER AND THE GEOMETRY OF THE
SURFACES.
1. Surfaces whose measure of curvature at every point is constant and nega-
tive were called pseudospherical surfaces by Beltrami in 1868, in order, as he
said, "to avoid circumlocution." Since, therefore, the definition of these surfaces
depends upon the definition of the measure of curvature itself, their history
may be considered as commencing in 1827, when Gauss ^ in his great memoir
entitled " Disquisitiones generales circa superficies curvas " establislied the idea
of curvature as it is understood today.
2. In this famous paper Gauss borrowed from the astronomers the notion
of spherical representation and established a point-to-point correspondence
between a curved surface and a sphere of unit radius. He supposed a ra-
dius of the sphere to be drawn parallel to the assumed positive direction of
the normal to the curved surface at a point /*, and the extremity of the
radius to be a point p corresponding to P of the surface. He defined the
total curvature of a part of the surface enclosed within certain limits as
the area of the figure on the sphere corresponding to it, and distinguished
this curvature from the very important notion of the measure of curvature of
the surface at a point, which is sometimes also called total curvature. This last
is defined as the quotient of the total curvature of the surface element at the
])oint by the area of the surface element, or in other words, " the ratio of the
infinitely small areas that correspond to one another on the curved surface and
on the sphere." He remarked further that " the position of a figure on the
sphere can be either similar to the position of the corresponding figure on the
curved surface or the inverse." When the position of two corresponding fig-
ures, the one on the surface, the other on the sphere, is similar, he called the
curved surface a convexo-convex surface, or a surface with positive curvature.
When the position of the figure is inverse to that of the figure on the surface he
called the surface a concavo-convex surface or a surface with negative curvature.
Gauss introduced various analytic expressions for this measure of curvature at
a point which he denoted by K^ among others, using a general parametric repre-
sentation of a surface through the parameters u, v, he found that
DB" - D'-
~ EG- F-'
where D, D\ D'\ E, F and G, functions of « and v, are tiie coefficients of
10 E. M. CODDINGTON.
the first and second fundamental differential expressions of the surface, the one
for the square of the linear element
ds" = Edu^ + 2Fdudv + Gdv-*
the other
- 1dxdX= Ddii? + 2D'dudv + D"dv\
where X, Y, Z are the direction cosines of a normal to the surface at a point
X, y, z.
When he chose as a special system of parametric lines a family of geodesic
lines and their orthogonal trajectories, he showed that E becomes a function of
u alone and F vanishes, so that the expression for the line element assumes the
form
dr = dii^ + Gdv^,
and he could derive for the curvature a simple corresponding form
1 dWG
VG Su? •
In particular he observed that if the system is a geodesic polar system in
which the u-curves proceed from a point and v is the angle that each geodesic
«-curve makes with an arbitrary but fixed I'-eurve and m is the arc-length of
each geodesic from the point, then G is a. function which satisfies the equations
("^'"«=»- i^^lr'-
and that if the u-curves form a geodesic parallel system, that is, if the v geo-
desies are orthogonal to a geodesic curve m = and v is as before the arc-length
along the v curves from m = and the arc-length v is measured on the curve
u = from some fixed point, the function G satisfies the equations
Lastly he wrote
(l+p' + q^f'
where the surface is represented by
z =f(x, y),
and J}, q, r, s, t have their usual meaning as the partial derivatives of z with
respect to x and y.
--(I)' -^l-|. "-Kfc')'.
da du' dv du du do' dv dv '
Bianchi'6», §§33, 46.
HISTORICAL DEVELOPMENT OP PSEUDOSPHERICAL SURFACES. 11
The measure of curvature at a point as defined by Gauss has now been
adopted as the standard definition and called the Gaussian measure of curvature
or total curvature. Both before and after the time of Gauss various definitions
of curvature of a surface had been advanced by Euler, Meusnier, Monge and
Dupin, but these definitions have not recommended themselves and are now
almost forgotten.
Gauss did not write directly on the subject of pseudospherical surfaces, but
in his memoir just quoted he published two important discoveries which were
afterwards easily applied to the special case of these surfaces. To Gauss is
due the celebrated theorem on the total curvature, (curvatura integra), of a geo-
desic triangle, for making use of the geodesic polar system he found that the
total curvature of a triangle whose angles are A, B , C is
A + B+ C'-TT
which is negative for surfaces of negative curvature and positive for surfaces of
positive curvature. An immediate inference from this and what may be
regarded as the first theorem in the geometry of pseudospherical surfaces is
that the area of a geodesic triangle on one of these surfaces is proportional to
its spherical deficiency. This theorem was proved by Bertrami in 1868.*
3. Gauss also established the well-known theorem that II is an invariant of
bending, that is, any disturbing of the shape of the surface which does not
involve stretching or crushing, leaves the value of ly at any point unaltei'ed.
Thus if one surface is applicable upon another the measure of curvature at coi'-
responding points of the two surfaces is the same. It was this invariant character
of A' that first gave interest to the study of surfaces of coustant curvature.
Gauss himself made no study of them, but Minding'^ in a paper of 1839, of
which more will shortly be said, discussed the sufficiency of Gauss' theorem for
the applicability of one surface upon another, and established that for surfaces
of constant curvature, and for these only, is Gauss' theorem a sufficient as well
as a necessary condition.
In a previous paper in 1830 he had integrated the Gaussian equation
^.__ 1 d'l/G
VG ^«' '
assuming Kto be constant, and had obtained the expression for the linear ele-
ment
When about to apply one surface upon another he accordingly wrote for the
expressions for their linear elements
* Page 37.
12 E. M. CODDINGTON.
f/s = (/m +1 -7^ — I dv ,
in which the primes indicate the elements with reference to the second surface.
The analytical condition of applicability
ds = ds'
he satisfied by putting
u = it\ V ^ a + v'
where a may have any value from zero to infinity. The first equation shows
that any jjoint on the first surface may be made to correspond with any point on
the second surface and the second equation that any geodesic curve on the first
surface proceeding from the point may be made to correspond with any geodesic
curve on the second surface proceeding from a corresponding point. Thus the
surfaces are applicable upon each other in oo' ways, or to quote IMinding, " One
can place two arbitrary points of the one upon two arbitrary points of the other,
provided that the lengths of the shortest lines upon the surface between the
pairs of points are equal to each other.'"
Becoming interested in the study of surfaces of constant curvature Minding^
proceeded to determine some of these surfaces. When the surface is assumed
to be of the form of z =f{x, y), the differential equation of his problem is
d^z d'z / c-z Y
dxJ^ dy^ \ dxdy )
where /r= a constant.
"This integration," he said, "has never been effected up to the present time
except for A'= 0." Changing the form of the equation by writing .r = r cosyjr
and y = r sin yjr in the place of x and y, he attempted its solution oulj- for the
special case
dz
— =z h ( /i ^ a constant ) .
A first integration gave him
dz = I>d^lr±l^l ^r-^-, -l-^]d>- (^,__ j,„„,t^,„t „i. i^te;;ration ,
*"^-(^^-i-p)<"- (5=;
For surfaces of constant negative curvature he first put a- = in this equation
of z and retained A : then he let It vanish and u" remain. When the first condi-
tion is fulfilled, he said, " the equation for s represents a curve which generates
HISTORICAL DEVELOPMENT OF PSEUDOSPHERICAL SURFACES. 13
the surface in the same way as the straight line generates the helicoidal surface
namely, by a revohitiou about the axis of :i while at the same time all its points
have a common motion parallel with 2." This surface was afterwards investi-
gated by Dini -*• ■' and called the Dini helicoidal surface. When the second
condition exists, Minding found that the surface becomes a surface of rotation
and the « curves become its meridian lines. He discovered three types of these
surfaces, those for which cr has a positive value, those for which it has a neg-
ative value and those for which it is equal to zero. It was probably his
original expression for the linear element
/sin (; — tanh d>.
cosh /
their values in terms of u and v, the parallel and meridian curves of the surface
to be deformed, he found three sets of equations for transforming a surface of
the first, second and third type respectively into a pseudosphere :
(1)
Ml
e
V e" = a smh -sm v,
- « / . 1 " , ''\
":= j^l Sinn — cos v + cosh I ■
K \ a a J
(2)
«1 g— M/a
6"=
■u, =
(3)
1—^2+ CiV-"^"
u^ =a log cosh — \- av
V, = a tanh e"
The first of these sets of equations is the same as the one given by Minding,*
if in the latter the point (1, 0) be chosen for the point (i', t). The third set is
identical with Dini's, for expressed in the same notation Dini's equations become
V = log tanh u — log v^, u^ + log i\ = log sinh u,
and when one is subtracted from the other,
u^ = log cosh u + V.
The validity of transforming the general expression for the linear element
ds- = Edit- + IFdudv + Gdv-
*Page 14.
22 E. M, CODDINGTON.
11 to the form in terms of the conjugate complex variables 6 and ^,
ds = 7 ^1 (« = curvature)
^-.y
was investigated by Weingarten" three years later. He decided that such a
transformation is permissible for surfaces of constant curvature, and that then
the reciprocal of the differential quotient dOjcv must satisfy two partial dif-
feiential equations.
The geometrical interpretation of the difference in the value of the constant
in the equation for the meridian curve of the three types of surfaces of rotation
was clearly determined by Beltrami.'- The early papers of this mathematician
on the subject of pseudospherical surfaces were contemporaneous with those of
Dini and both appeared side by side in the Italian journals during the years
1864 and 1865.
Beltrami-" wrote a long treatise on the general theory of surfaces in which
he considered in particular geodesic curvature, evolute and involute surfaces and
differential parameters. He applied the various theorems that he obtained to
the special case of pseudospherical surfaces, all of which theorems will be dis-
cussed in detail in the chapter on Evolute Surfaces and tlie Transformation
Theory. In the same year that this treatise appeared Beltrami -' published a
paj^er devoted entirely to the pseudosphere in which he investigated its geo-
metrical jjroperties.
He gave a geometrical proof of Dini's statement that the second class of
pseudospherical surfaces of rotation is composed of a single surface, which is
equivalent to saying that the pseudosphere is always bent into a pseudosphere,
that is, it is identical with itself, if bent when the meridians are retained as
meridians. He first showed that the radius of geodesic curvature of everj'
parallel of a pseudosphere, being equal to the length of the tangent to the
meridian curve between its point of tangencj' and the axis of revolution, is a con-
stant R where —l/H' is the total curvature of the surface. Having proved
that geodesic curvature is an invariant of bending, he then observed that, when
the pseudosphere is so deformed that its meridian curves remain meridian curves,
the radius of a geodesic curvature of every parallel circle of the deformed surface
is JR, that its meridians are therefore curves with tangents of constant length,
and that they must therefore be identical in form with the meridians of the
original surface, since to one value for the tangent length there corresponds but
one tractrix.
He also showed, at this time, that the area and volume of a pseudosphere are
equal to those of a sphere with the same numerical value of curvature, thus
making an analogy between the simplest forms of surfaces of rotation with con-
stant positive curvature and constant negative curvature.
HISTORICAL DEVELOPMENT OF PSEUDOSPHERICAL SURFACES. 23
Belti-ami's ^- most important paper in regard to surfaces with constant nega-
tive curvature was published in 1868, under the title of an Essay on the Inter-
pretation of the Non-Euclidean Geometry. Nothing written on the subject
since Minding's" paper in 1839 can be compared in importance with this cele
brated memoir by Beltrami, in which for the fii-st time was made clear the
relation of pseudospherical geometry to the general theory of geometry. The
contents of Beltrami's paper will be given later ; here attention only will be
called to the fact that, although the greater part of the essay is devoted to the
demonstration of geometrical propositions, yet it contains a proof of his discovery
that the difference in the expression for the linear element of the surface of rota-
tion results from the difference in the nature of the parallel circles chosen for
one system of parameters, that the centers of those circles may be real finite
points, points at infinity or imaginary points, and that consequently the corre-
sjjonding surface will be of the first, second or third form given by Dini.
To return to the development of the helicoidal surface with constant negative
curvature it will be remembered that Dini -^' "' made a special study of this sur-
face. He first communicated his results to the French Academy iu 18G5, and
there stated that the surface is generated by a tractrix moving along a helix
that lies on a cylinder. Recalling Bour's'" Theorem, that helicoidal surfaces are
applicable upon surfaces of rotation, he divided them into two classes, according
to whether they are applicable upon a sphere or a pseudospherical surface. He
used the ordinary expressions for a point on a helicoidal surface,
X = u cos V , ?/ = u sin v, j; = niv + <^ ( m )
where rn multiplied by Stt is the rise of the helix and 0(m) is a function that
determines the form of the generating profile. He found that this generating
profile (^ ( w ) must satisfy the differential equation
in the case of surfaces of negative constant curvature — l/cr. This he reduced to
„.[i +(:*)= ]=»=-„.,
by putting Ijm' for 1Z-.
Since the left hand member of this last equation is the square of the length
of the tangent to the generating curve between the axis of revolution and its
point of contact and the right hand member is a constant, Dini thus obtained
an infinity of new helicoidal surfaces of negative curvature — l/o", each cor-
responding to a value of m and generated by a tractrix of tangent length
Va^ — rr? moving about a cylinder and all developable upon the same
pseudosphere.
24 E. M. CODDINGTON.
5. Up to 1868 the ouly surfaces of constant negative curvature that had been
determined and studied were surfaces of rotation and helicoidal surfaces. To
these were added a new group of surfaces by Enneper '''' ■*" in 1868. In a
memoir written in that year he determined all the surfaces of constant curva-
ture, one of whose families of lines of curvature is composed of plane curves or
of spherical curves. As he showed, the presence of a system of plane lines of
curvature on a surface of constant curvature requires that the second system of
lines of curvature should be spherical, and conversely, moreover, the planes of
the oue system meet in a straight line and the centers of the spheres of the other
systems lie on that line. The surfaces possessing these characteristics, whether
of positive or negative constant curvature, have since been called Enneper's
surfaces.
The determination of surfaces with either plane or spherical lines of curva-
ture had already been attacked by Joachimsthal, Bonnet and Serret, and Bonnet'
in particular examined surfaces for which the lines of curvature of the one
family are plane and those of the other family are spherical, and showed that
for surfaces of rotation " the lines of the one system are in planes all of which
pass through the same straight line and the centres of the spheres on which are
traced the spherical lines of curvature can lie on a right line."
Enneper considered surfaces of constant curvature not of rotation and set
1 1
assuming u and v as the parameters of the lines of curvature of which it = a
constant are the plane lines of curvature and R^ and R^ are the principal radii
of curvature. Putting
1 1 1 - ii 1 1 1 + <
and employing the equations
d V^ 1 dVE j^'^ __^ ^y"^
dv R^ ~R'^ ~W ' d^tlf^^li^ ~a^ '
he obtained the following equation in t
, . d- tan-' t d tan-^ t 1 — t^ 1
and the equations for U and G,
Denoting by a the angle the plane u = a constant makes with the surface, since
I
HISTORICAL DEVELOPMENT OF P8EUD08PHERICAL SURFACES. 25
VEG Su \ li, ,
he substituted the new variable
R.R. d (\/G\ r • , ,
(4) -— — 5- I —^f=r- I = — etn 0- = a tuuctiou of u alone,
'■ = 11
ctn crdu .
He obtained from (1), (3) and (4) a differential erjuatiou for t. Integrating
this equation and substituting the value thus obtained for t in equation (2), he
differentiated the resulting equation twice with respect to ii and arrived at a
differential equation for v. The integral of this equation
^^— = A cosh 2?/, + B sinh 2m, + C,
contains three arbitrary constants. The form of the surface depends upon the
value of these constants and the relation that exists between them.
Enneper concluded there was no loss of generality, in putting B = , and
investigated accordingly. Later Kuen ^ showed that thereby a class of surfaces
was overlooked.
After considerable reductions in which the two equations,
/ dv Y
I jr y-' I =C — A cosh 2rj {i\ = function of v alone) ,
formed a chief element, he proved that the planes of the lines of curvature 71 =
a constant meet in a straight line, and that the lines of curvature r = a constant
lie on spheres that cut the surface orthogonally and whose centers lie on the
straight line.
In choosing the fixed line as the axis of z and representing by (/> the angle
between the intersection of the plane with the xy plane and the axis of x, he
defined the surface by the following three equations,
X sin 4> — y cos ^ = ,
/ . • , . <14> sin cr
( X cos d) + wsina))-^- = , ^ r ,
V r-ry r; ^^ cosh(M, + «,)'
- i (C' — A cosh 2 V, ) dv — r/ tanh ( «, + v, ) 'l'^ = ''^
'J J ' ' •' ^ ^ ^' dv \
where <^ satisfies the equation
/#\2 C^-A^ . ,
Vdu)^-^^'^-'^-
26 E. M. CODDINGTON.
A detailed study of Enneper's surfaces was made by Bockwoldt '^' in 1874
for surfaces of constant positive curvature and by Lenz ^^ in 1879 for surfaces
of constant negative curvature in which the coordinates of points on the surface
are expressed in elliptic functions of the two parameters.
There is one case and it was considered by Enneper in which the surface is
expressed through the elementary functions, namely, when a- is constant and B
as before is zero. He then took
and the equation of the surface becomes
z= g cos (7 tan~' — h V g''' sin^ a — x^ — y^
X
, . , / ff sin o- + i/ <7" sin^ a- — x^ — y'\
-iff sin a- log I —, 7 ^ . ^ 2 2 ) '
\c/ sma-— y g^ sin" a- — ar — y'' /
an equation which shows that the surface is generated by a tractrix whose ver-
tex describes a helix on a right circular cylinder. This is Dini's helicoidal sur-
face and it is thus found to occupy a special position in the Enneper's surfaces.
Kuen*'^ in an interesting paper in 1884 set forth in a clear light the relations
between the surfaces determined by Enneper and those which could be derived
from the three surfaces of rotation by Bianchi's method for deriving one pseudo-
spherical surface from another when the geodesic curves with reference to which
they are derived meet at a point at infinity.*
6. Geodesic lines and their orthogonal trajectories were the only curves con-
sidered on pseudosj^herical surfaces until 1870 when Enneper"' began to write
on asymptotic curves. Since real asymptotic curves cannot exist except on sur-
faces of negative curvature, Enneper began his investigations on the subject for
these surfaces only, afterwards supposing the curvature to be constant as well
as negative. Asymptotic curves on pseudospherical surfaces possess peculiar
properties that render them important in the infinitesimal deformation of a sur-
face and in the deriving of a new pseudospherical surface from one that is
known. For the latter operation it is important to know the expression for the
linear element of the surface referred to asymptotic curves. Enneper found
this expression by inserting in the Codazzi-Mainardi fundamental equations
d / D \ a / D' \ dv -^ du dv B
* Page 56.
HISTORICAL DEVELOPMENT OF PSEUDOSPHERICAL SURFACES. 27
dn dv D \ vu da dv J D
d / D" \ d ( D' \ ^ dv dv ^ du D
i B" \ d / Z)' \ ^ dv ^ dv ^
^u\^EG-F-} ^^WEG-F-I 2{EG-F-) VEG-F'
[EG-F-') VEG-F' ^ 2{EG-F') VEG-F'
the values D = 0, D' = and A^= — l/R- = a constant, which are the condi-
tions that the lines u = a constant and v = a constant shall be asymptotic
curves on a surface with curvature — 1/i?". He thus reduced the equations to
do du du dv
from which he saw that jB" is a function of u alone and G^ is a function of v
alone, so that the expression for the linear element may be written
ds" = chr -f 2Fdudv + dv'
and the equation for the curvature A' becomes
1 _ 1 5^2m
~ i?- ~ sin 2(u du dv
where 2a> = the angle between the asymptotic curves and F = cos 2a).
Enneper discovered the famous theorem known as Enneper's theorem, that
the square of the radius of torsion of an asymptotic curve at every point is equal
to the product of the principal radii of curvature of the surface at that point,
with the minus sign placed before it. In proving this theorem he obtained the
two following equations for the curvature l/p„ and the torsion l/i\ of the
asymptotic curve, v = a. constant,
1 cmXy"^) ~dV 1 I)'
P„ VEG-F' '•„ VEG-F'
the first of which shows that its geodesic curvature is equal to its curvature,
and the second that its torsion squared is equal in value but opposite iu sign to
the curvature of the surface, and is consequently constant when the curvature of
the surface is constant.
Enneper also remarked that if one surface is applied on another, one sj'stem
28 E. M. CODDINGTON.
only of asymptotic curves on the first surface can by any possibility pass over
into asymptotic curves on the second surface, as, for example, the generators of
a skew surface.
In the same year, Dini^' made a study of aymptotic curves. Supposing the
surface to be represented upon a sphere after the method of Gauss, he denoted
its linear element referred to arbitrary parameters m and v by
els- = Bchr + 2Fchi dv + Gdv\
and the spherical image of its linear element by
ds- = B'du- + IF'dudv + G'dv\
He derived the Codazzi-Mainardi equations for the coefficients E', F', G', and
for the coefficients D, D', D", of the second fundamental differential expres-
sion for the surface and introducing the conditions necessary in order that the
parametric lines on the sphere should represent the asymptotic curves of a sur-
face of negative curvature — A^", he reduced these equations to
F' G'
dv ou \fi J ou cv\iM J
and to
dv ~ ' du
vehen /a is a constant.
His expression for the spherical representation of the linear element is there-
fore,
ds'- = dir + 2F' dudv + dv^
and for the linear element of the surface itself it is
dir — 2F'dttdv + dv-
ds- = o ,
for in an earlier paper he had remarked that the arc lengths of asymptotic curves
are always proportional to the arc lengths of their spherical image in the i-atio
of the curvature of the sphere to the curvature of the surface and that the angle
between two asymptotic curves on the surface is the supplement of the angle
between the two lines that represent them on the sphere.
Dini was the first to observe from the form of this expression that " asymp-
totic curves divide a surface into infinitely small lozenges." Hazzidakis^' went
a step further than Dini and found the area of one of these lozenges to be
A + B + C + D — 2- where A, B, C, D represent its four angles. He
obtained this value by integrating along its bouudarj' the equation for the area
of the quadrilateral
HISTORICAL DEVELOPMENT OF PSEUDOSPHERICAL SURFACES. 29
r fsin 2eochi dv ,
where sin 2(» is given by the equation for the measure of curvature IT,
sin 2a) du dv
Voss"' approached the subject of asymptotic curves from the consideration of
equi-distant curves. He gave that name to a system of curves on a surface which
form a net-work of quadrilaterals whose ojiposite side are equal. His expres-
sion for the linear element of the surface, when u = a constant and v = a con-
stant represent these lines, becomes
ds'- = (hr + 2 cos icodit dv -f dv-,
where 2a> = the obtuse angle of a quadrilateral. To find the surface for which
the equi-distant curves are asymptotic lines he made the necessary substitutions
in the Codazzi-Mainardi equations
Z»=0, Z>" = 0,
^=1, G = l,
and reduced them to two partial differential equations
whose common solution is
D'
^^ = a constant.
i/T
This expression denotes the measure of curvature of the surface with the nega-
tive sign, so that Voss thus proved that only upon surfaces of constant negative
curvature can a system of equi-distant curves be composed of asymptotic lines.
Voss also found the characteristic equation for surfaces of constant negative
curvature — \ j R~
0-2(0 sin 2a)
dudv R-
by deforming the meridian and parallel curves of the pseudosphere into equi-
distant curves.
The equation for an asymptotic curve on a pseudosphere was derived by
Beltrami^' in 1872. In that year he published a paper whose title, "On the
Surface of Rotation that serves as a Type for all Pseudospherical Surfaces,"
shows the nature of its contents. In order to obtain a set of equations for the
surface he called the axis of rotation the axis of s, the plane of the maxinuim
parallel circle the x y plane, the angle measured on this plane that any meridian
30 E. M. CODDINGTON.
makes with a fixed meridian the angle cj) and the acute angle that the tangent
to the meridian at anj- point makes with the axis of rotation the angle 6. His
equations for the coordinates x,y,zoi a point on the surface of curvature — l/?-"
are then
0'
: r sin 6 cos <^, y =r sin ^ sin ^, z = r I log ctn „ — cos 6 y
Every point on the surface is therefore determined by the value of and 6 at
that point. The expression for the linear element of the surface then becomes
ds- = r' ( ctn= 6(10- + sin= 6 d4>'' )
and may be transformed into the usual form
ds- =cht^ + r^ €-""''■ dv-
by means of the equations
sin^=e-"'',
T — ~ gy ds~ ds
and when the curve is an asymptotic cuive 1^ = 0, the arc length of an asymp-
totic curve is given by
HISTORICAL DEVELOPMENT OF PSEUDOSPHERICAL SURFACES. 31
ds = ± rd = 4> and 4> = (f>^
is equal to its orthogonal projection on the plane of the maximum geodesic
circle and that each one of the infinite number of portions into which the length
of the curve is divided by a meridian curve is equal to the circumference of the
maximum parallel. If a linear element of surface be taken along an asymptotic
curve, there results the equation,
ds' = r- ( ctu= Odd- + sin= Bdcj}- ) = r^ d^
or
:^J^de = d4>
sm p
and the integral of this equation,
log tan 2 = - ^) = 1 ,
gives the equation of an asymptotic curve that touches the maximum parallel at
the point „.
The part played by an asymptotic line in the infinitesimal deformation of sur-
faces was discovered by Jellet'- in 1853, who gave the theorem that, when a sur-
face is deformed infinitely little, one asymptotic curve may remain unchanged.
He therefore called asymptotic lines "curves of flexion," and stated the proposi-
tion, " We can fix a curve of flexion without preventing the deformation of any
finite portion of the surface." As only surfaces of negative curvature have real
asymptotic curves, they are the only surfaces that can be bent while a curve on
them is unchanged. This theorem was demonstrated by Lecornu'^^ in 1880 and
by Weingarten'^ in 1886.
In this connection, Weingarten introduced the idea of the bending invariant.
He denoted by ex, ey, ez the infinitesimal increments that each coordinate
X, y/, 2 of a point receives when the surface on which it lies is bent infinitely
little, and by x' , y\ z the coordinates of the same point after the surface is
bent so that
x' = X 4- ex, 7/' =y + ey, z = z + ez.
Then assuming the expression for the linear element of the surface to be
ds' = Bdu' -f 2Fdu dv + Gdv;
he defined the bending invariant <^ by means of the equation
32 E. M. CODDINGTON.
1 /^,5i dx -^dx dx\
'P — ~ yEG — F- \ ^M dv~ ^ ~dv du) '
He, moreover, showed that since the linear element of the surface remains
unchanged when the surface is bent, and e is so small that when raised to the
second power it may be neglected, it must happen that
2cZa; dS = ,
ov X, y,z may be regarded as the coordinates of a point on the second surface
that corresponds to the first by the orthogonality of its elements.
A second theorem of bending that relates to surfaces of negative curvature is
that when two surfaces are associated, that is, when the bending invariant of the
one at every point is equal to the distance of the tangential plane at correspond-
ing points of the other from the origin, the total curvature of one of the sur-
faces must be negative. The discovery both of the existence of such pairs of
surfaces and of the theorem concerning them is the work of Bianchi."'* The
associated surface of pseudospherical surfaces have been studied within the
past few yeai-s by Cosserat,'^"' "' Guichard '"■ "' and Voss.
The use of asymptotic lines in the transformation of one pseudospherical sur-
face into another will be considered later. It is necessary here to turn to the
development of the geometry of lines on these surfaces.
7. "While the mathematicians of France, Italy and Germany were discovering
the various properties of surfaces of negative constant curvature and adding
from time to time to the development of their theory so as to make them take
an important position in the class of surfaces of constant curvature, considering
them merely as a necessary adjunct to the completion of the study of that class,
another topic of more general interest was attracting the attention of men all
over the world. This matter was none other than the recognition that Euclid's
fifth postulate, equivalent to the statement that only one line parallel to a fixed
line can be drawn through a point, is not capable of demonstration from the pre-
ceding hypotheses.
Gauss ^ among others gave some study to the subject and recognized in con-
nection with it the existence of a new geometry, which he caUed the non-
Euclidian, and which he distinguished from the Euclidian by the essential char-
acteristic that in it there is never any similitude in the figures without equality.
Gauss never published a complete exposition of his theories, but referred to
them occasionally in various papers since published in the Gottingen edition of
his works and in his correspondence with Schumacher. It is in a letter to the
latter that he gave the now familiar expression for the semi-circumference of a
circle with radius r in the non-Euclidian geometry,
■n-K ^
-— — (e'^ e ^) ( JSr= a constant )
HISTORICAL DEVELOPMENT OF PSEUDOSPHERICAL SURFACES. 33
and remarked that for the Euclidian geometry K becomes infinitely great, but
Gauss' contributions to the new geometry were slight in comparison to those of
Lobatchewsky.
In 1831 Lobatchewsky* produced a pamphlet on the theory of parallel lines,
of which Gauss said in another letter to Schumacher, " I have found in the work
of Lobatchewsk}^ no surface new to me, but the statement is entirely different
from that I had contemplated."
In this pamphlet, Lobatchewsky set forth a whole imaginary geometry based
on the hypothesis that two lines parallel to a third may be drawn through a
point and demonstrated a series of jjropositions analogous to those of the Euclid-
ian geometry. In 1854 Riemann'^ wrote his renowned Habilitationschrift in
which he introduced for byperspace of any dimensions the idea of three kinds of
constant curvature, positive, zero and negative. In particular he considered
two-fold space and stated that all surfaces of positive curvature are developable
upon a sphere and all those of zero curvature upon a cylinder. " Surfaces of
negative curvature,"' he said, " will touch the cylinder externally and be found
like the inner position (towards the axis) of the surface of a ring." He made
the further statement that " the surfaces with positive curvature can always be
so formed that figures may also be moved arbitrarily about upon them without
bending, namely they may be formed into sphere surfaces : but not those with
negative curvature." He thus suggested the idea of a geometry on surfaces of
constant negative curvature as opposed to spherical and plane Euclidian geom-
etry. No mathematician, however, connected the new geometry of Lobatchewskj'
with the geometry of pseudospherical surfaces until Beltrami '- wrote his essay
on the non-Euclidian geometry in which he showed analytically that all the
propositions and theorems of the new geometry can be realized by means of
figures lying upon such a surface.
His method of proof was based upon such a choice of parameters u and v that
a linear equation between them represents a geodesic line on the surface. Con-
sequently the surface may be represented upon a plane in such a way that its
geodesic lines become straight lines. Beltrami -* wrote a paper in 1865 on this
representation showing that it is possible only for surfaces with constant curva-
ture, and that it is analogous to the central projection of a sphere together with
its linear substitutions.
He rej)resented the geodesic lines by
au + &y + c =
and the straight line on the plane by
ax -\-hii + c=^
so that the equations for transforming the one into the other are
34 E. Jf. CODDINGTON.
u = x, v = y,
and the plane and surface correspond to each other point to point.
Since the general differential equation of any geodesic curve is
2{dud'-v — d'udv)= 2(EG — F')
f/ BE dF d_E\ ( DG dE BE d F\
[F-p-iE ^+E^\du^-{2E ^ -G -^- -ZF -^+1F^ \dirdv
\ cu cu ov ) \ cu cu vv cu )
( dE dG dG dF\ ( dF BG BG\
-i-2G^+E^+^F^-2F^]dudv'+[2G^-F~-G^yM
\ Bv ov Bu Bv J \ Bv Bv Bu )
and in the case considered this must become
du d'v — dhi dv =: ,
he saw that the coefficient of each term of the right-hand member of the equa-
tion must vanish identically so that the reduction furnishes a set of equations
whose integrals give for E, F and G the values
(m^ ^. ^2 + arf ~ {u- + v- + a')-' ~ (?r + v- + a^f
where l/H- is the curvature of the surface and a is an arbitrary constant.
He was thus able to write down at once the expression for the linear element
of a surface with constant positive curvature
-S- ( ( rr + 1)^ ) dii^ — 2uvdu do -f ( a" -f ir ) dv' \
(ir + V + a-y
which he changed into
i?" ((«' — v-)dir + 'luvdudv + («- — u')dv- )
ds' =
for pseudospherical surfaces by wi-iting — H' and — a' for S' and a'.
This expression for the linear element was his starting point for his investi-
gations on the non-Euclidian geometry in 1868 and from it be developed other
properties peculiar to the surface and to Lobatchewsky's imaginary plane. He
observed that if be the angle between two lines u = a constant and v = a
constant that
uv . ^ aVa' — u^ — v'
cos ff = , sm ^ =
By using polar coordinates r and (f> he found a second expression for the linear
element for the surface
HISTORICAL DEVELOPMENT OF rSEUDOSPHERICAL SURFACES. 35
* = ^'|(^0'-^^=1'
From this he derived equations for the length /o of a geodesic line (^ = a con-
stant and for the arc o- of a geodesic circle ;■ = a constant, or as he called it, a
geodesic circumference,
B , a+\/v? + v^ R^ a+r
)• = a tauh
p
a = <^R sinh j^ .
His expression for the circumference of a geodesic circle is therefore similar to
to the one found by Gauss,
7r^(e'=''^-e-''>').
From these equations he saw that the curve whose equation is
u' + V' = «',
bounds the region of real values. He remarked that when the surface is repre-
sented upon a plane this curve becomes a circle which he called the limiting
circle because all the points corresponding to real points on the surface lie
within it, all those corresponding to the ideal or imaginary points on the surface
lie without it and the points on its circumference correspond to infinitely far off
points on the surface. He also showed that the geodesic lines of the surface
become chords of this circle and that, since two points fully determine a chord,
two points will determine a geodesic line.
From the equation for d he further observed the nature of the parametric lines
on the surface, that they consist of two systems of geodesic lines which ai-e so
related to each other that the two fundamental lines u = 0, u = 0, meet at right
angles at the origin while the coordinate lines u = a constant are orthogonal to
V = and the coordinate lines v —- a constant are orthogonal to ?f = .
He showed by a rigorous proof that any two lines that cut each other orthog-
onally may be chosen for the fundamental lines, instead of m = , v = , and
that consequently any geodesic line may be made to coincide with any other and
the surface superposed upon itself, for changing the pair of orthogonal geo-
desic lines intersecting at the origin into any other set of orthogonal geodesic
lines through any other point does not alter the form of the expression for the
linear element.
These two characteristics, the superposability of the surface upon itself and
the determination of a geodesic line by two points, Beltrami called the " funda-
36 E. M. CODDINGTON.
mental criteria of elementary geometry," and since they belong equally to pseiido-
spberieal surfaces and to the Lobatchewskian plane, he said : " It becomes
evident that the theorems of the plane non-Euclidian geometry exist uncon-
ditionally for all the surfaces of constant negative curvature."
The keystone of the non-Euclidian geometry is the proposition that two
straight lines can be drawn through any fixed point parallel to a given straight
line. Beltrami proved this proposition by means of his geodesic representation
of the pseudospherical surface on the plane in the following manner: first it is
necessary to show that the angle between two geodesic lines intersecting at a
real finite point on the surface is never nor tt, but that the angle may be
or TT when the curves intersect at a point of infinity. If this angle is repre-
sented by yfr and the angle on the plane between two chords corresponding to the
geodesic lines be represented by ifr' and the angles which the chords make with
the axis of ^Y by /m and v respectively, -\]r and -yjr' are related by means of the
equation
a{\/a^ — u^ — v^) sin -v/r'
tan i/r
' cos 1^' — ( w cos /A — u sin /^ ) ( t' cos v — u sin v ) '
the right-hand member of which can only be zero when ?r -)- v" = «", that is
when the two chords meet on the perimeter of the circle, consequently the angle
i/r is only when the geodesic curves meet at a point at infinity which corresponds
with the point of intersection of the chords on the perimeter of the limiting
circle.
When a given geodesic and a given point on the surface are represented by a
chord of the limiting circle on the plane and a point within its perimeter, two
of the chords drawn through that point will meet the first chord at its extremi-
ties on the circumference of the circle, therefore, the two geodesic lines which
correspond to the two chords will meet the given geodesic line at infinity, mak-
ing with it an angle and they will therefore both be parallel to it though
drawTi through one point.
Following the line of thought laid down by Lobatchewsky, Beltrami next
defined the angle of parallelism 11 as half the angle between the two geodesic
lines drawn through a fixed point parallel to a given geodesic line. To deter-
mine tan n he constructed the corresponding angle and lines on the plane. He
chose the center of the limiting circle to represent the fixed point and the line
corresponding to the geodesic bisector of the angle of parallelism for the axis of
X so that the coordinates of the extremities of the chord representing the given
geodesic line are (x, y) and {x, — y). He could then write
y Va'-x'
tan II = =
and from the equation
HISTORICAL DEVELOPMENT OF PSEUDOSPHERICAL SURFACES. 37
r = \^u^ -\- v^-=a tanh -^ (p = length of geodesic bisector),
he could obtain for x along the axis, y = 0, the value
o
x = a tanh ^51
Ji
so that on the surface
1
tan n =
sinh ^
a form equivalent to the one given by Lobatchewsky.
By means of the above equation he was able to express Minding's equation
for the angle of a geodesic triangle in terms of the angles of parallelism of the
sides and thus obtain the fundamental equation of the non-Euclidian trigonometry
sin n (6) sin n (c) ,
cos A cos n (b) cos n (c + Mt- / n -^
^ ^ ^ ^ sin n (a)
where a, b, c are the sides and A, B, C the angles of the triangle.
Finally he found the area of a triangle to be proportional to its spherical
deficiency, a fact which results from Gauss" theorem that its total curvature is
equal to the sum of its angles minus tt.
From the theorems of pure geometry Beltrami returned to the subject of the
three different forms of the pseudospherical surface of rotation. He first found
the equation for a geodesic circle, or as he called it a geodesic circumference,
whose center is the point u, v, and whose radius is p, to be
V(a^ — II' — v^) (a- — ?/j — ■«,";) -"'
and by means of it he deduced that the expression for the linear element assumes
one of the three different forms given by Dini,
ds' = ((hr + sinh" -^ dv'),
ds' = {dir + e-^'i^dv'),
ds' = (du'^ + cosh^ p Jr),
according as to whether the centers of the geodesic circumferences chosen for
one family of parametric curves are real, at infinity or ideal, that is whether on
the plane, the corresponding points lie within the limiting circle, on its perimeter
or entirely without it.
He also remarked upon the peculiar properties of the three types of geodesic
453:173
38 E. M. CODDINGTON.
circumferences, how, those of the third tj'pe with a common center are parallel
to a geodesic curve, a property which belongs to all geodesic circles on a sphere,
but which belongs only to geodesic circles with an ideal center on a pseudospheri-
cal surface ; how a geodesic circle of the second type is identical with what is
known as the limiting circle or horicycle of Lobatchewsky, that is, a curved
line such that all the perpendiculars erected at the middle point of its chords are
parallel to each other ; how the geodesic lines orthogonal to a family of geodesic
circles of the first type go through a common point usually chosen for the origin
{n = v=0).
In this same paper, in speaking of the three types of surfaces of rotation
which correspond to the above three forms of the linear element, he remarked
that in the actual application of a surface of rotation of the first type upon a
pseudospherical surface of a different form, it is necessary to make a slit in the
surface from the point of intersection ( ?< = v = ) of the meridian curves in
order to apply the " pseudospherical cap " about the point (^u = v = 0) upon
the second surface. He went on to observe that surfaces of rotation of the sec-
ond or third type have each a minimum parallel circle, that for the last named
surface, this minimum circle is the geodesic curve to which all the other par-
allel circles are parallel and that at equal distances from it, on either side, lie two
maximum parallel circles between which lies the real part of the surface and that
when a pseudospherical surface is applied upon a surface of rotation of the sec-
ond or third type it may be wrapped about it an infinite number of times. These
properties though evident from the drawings of pseudospherical surfaces at the
end of Mindiugs memoir in volume XIX of Crelles Journal, were not de-
scribed by him nor were they spoken of in any of the papers previously mentioned.
8. Beltrami^' wrote a paper in 1872 devoted exclusively to the subject of the
pseudosphere, making therein a particular study of its asymptotic and geodesic
curves. His theorems on geodesic lines, following in natural order after his
remarks in regard to these curves in his " Essay on the Interpretation of the
Non-Euclidian Geometry," will now be considered.
If the expression for the linear element of the pseudosphere of curvature
— 1 /}•■ be written in the usual form
ch- = (la- + e-"""!'- d4>^
where the parameter <7 represents the arc length of any meridian curve and the
parameter (^ denotes the angle that that meridian makes with a fixed meridian
measured on the plane of the maximum parallel, the general equation for the
radius of geodesic curvature of a curve becomes
HISTORICAL DEVELOPMENT OF PSEUDOSPHERICAL SURFACES. 39
Beltrami obtained the equation for a parallel circle.
by putting p = a constant in this equation and then integrating. The denomi-
nator set equal to zero gave him the differential equation for a geodesic curve
whose integral is
e'"'' + ((j) — Ij)- = (a + b-) . (n, 6, c, f7 are constants. )
If the equation for a parallel circle be differentiated twice with respect to it
will become
d'{e"''i'+cj>')_ hd-e^"!'
d^^ ~ a d^
and the combination of this equation with the equation for the radius of geo-
desic curvature makes the latter assume the form
[-(**)7
ar d^e'^i''
The comparison of this equation with the ordinary one of Differential Calculus
for the radius of curvature of a plane curve showed him that the two become
identical ii (f) = x and e"''' = y. He saw, therefore, that the geodesic circles of
the surface may be transformed into circles on a plane whose equation is
y- + e^ — ad h-p-
cr a"r- '
(^+iy+(--+«y^
and that the condition that these circles should have a center that is a real finite
point, a point at infinity or an imaginary point may be expressed analytically by
P = r.
This projection, which is similar to the stereographic projection of a sphere,
has since become very useful in the investigation of pseudospherical surfaces.
Beltrami's first method of projection or geodesic representation of a pseudo-
spherical surface converts the geodesic lines of a surface into straight lines on
the plane ; his second method transforms the geodesic lines on the surface into
circles on the plane.
Beltrami's second method of projection is conformal, and Busse"^ has shown
in a recent doctor's dissertation that it is only surfaces of constant curvature
that cau be conformally projected upon a plane in such a way that their geodesic
curves become right lines or the arcs of cii-cles.
A very interesting deduction was made by Cayley*" in 1884 from the theo-
rems contained in Beltrami's two papers on pseudospherical geometry, namely.
40 E. M. CODDINGTON.
that " the Lobatchewskian geometry is a geometry such as that of the imaginary
spherical surface ^Y- + Y- + Z"= — 1 (spoken of by Dini, page 18) and that
the imaginary surface may be bent without extension or contraction into the real
surface considered by Beltrami."
He remarked that this bending is an " imaginary process " for the points and
lines on the first surface are imaginary and those on the second are real, while
the angles and distances are real on both surfaces. He denoted the coordinates
of a point on the imaginary sphere of curvature — 1 by Jl, Y, Z , and the
coordinates of a point on a pseudosphere of the same curvature by x, y, z. He
was then able to transfer the linear element on the surface denoted by
ds- = dX'+ dY- + dZ'
into the linear elements of the pseudosphere represented by
ds" = dx" + dy- + dz",
by means of the three sets of equations
X^ .. -\ ., Y=
Vl — M^ — U^ ' 1 1
w
■ A J- — M
l/l-u'-v' ' ^ 1-m'
X = cos (j) sin 6 , y = sin ^ sin Q , 3 = log ctn ^ — cos 6 ,
where u and v are Beltrami's parameters which, when linearly connected, repre-
sent a geodesic curve.
Having established the fact that the imaginary sphere is transformable into a
real pseudosphere, Cayley proceeded to consider the geodesic curves on the first
surface. The equation of a geodesic curve on the imaginary sphere may be
written in the form similar to that of a geodesic curve on a real sphere
«A'+ bY -\- cZ =^ (o, 6, c, = constants) ,
but Cayley observed that " since for a point corresponding to a real point of a
pseudosphere ^ is a pure imaginary, and Y and Z are real, we see that for a
geodesic corresponding to a real geodesic of the pseudosphere X must be a
pure imaginary and X and Z real. In order to have aU the coefficients real he
therefoi'e made the substitutions
P = iX - Y, Q= iX + Y,
by means of which the equation becomes
( _ iia - ii)P + ( _ J-,-a + ih)Q + cZ=0,
or
AP + BQ+ CZ=0,
HISTORICAL DEVELOPMENT OF PSEUDOSPHEKICAL SURFACES. 41
where A , B and C are all real. Applying the same equations of transforma-
tion to this equation as he had formerly applied to the imaginary sphere to
deform it into a pseudosphere he found that the equation assumes the form
A + B{f"-^ + 4>') +(? = is the equation of the absolute in homogeneous line coordinates.
He saw that both expressions under the sigu of the logarithm are anharmonic
ratios, the first formed of four points, the second of four lines, each of which,
according to Cayley, belongs to a system in involution, and that therefore every
point in a line except the points of its intersection with the absolute may be
linearlj' transformed in every other point and every ray in a pencil, except the
two tangent to the absolute, may be linearly transformed into every other ray.
Klein investigated the nature of the absolute and discovered its characteristic
properties; first, that since for an imaginary value of C it is imaginary, for a
44 E. M. CODDINGTON.
real value of C it must be real, and that, in that case, since only real distances
are considered, the auharmonic ratio is positive and all real points lie within its
circumference ; second, that it lies at infinity, for if a conic is assumed to be a
circle with x as its center and y a point on its circumference, its radius will be
by Cayley's formula equal to
2/ecos-J^,
JX yy
and will become infinitely great when y lies on the circumference of the conic,
0^=0; third, that it is impossible to determine the region outside of the abso-
lute, " the ideal region," for by means of a linear transformation a man starting
from any point within the conic to walk to its infinitely far-off circumference at
a uniform velocity will never reach it, much less then will he know what lies
beyond.
He therefore concerned himself only with the points and angles within the
absolute and saw that for every line the fundamental elements are real, but that
for each pencil of rays they are imaginary, since no real tangent can be drawn
from an interior point to the conic. He then put for C the value i'/2, so that
the sum of the angles about a point is the same as in ordinary plane geometr}'.
This description of the absolute, that it is a real circle at infinity within whose
circumference lie all real points, is exactlj' the same as the definition of Belt-
rami's limiting circle, and Beltrami's expression
R ^ a + Vu"- + v"
-^ f — 1 tr + v^
for the length of a geodesic line from the center of this circle is exactly the same
as Klein's, if C = Ji/2. Consequently the propositions proved by Beltrami with
respect to parallel lines, the angle of parallelism and trigonometrical ratios belong
equally to the metrical geometry and may be solved by means of figures drawn
in the plane of the absolute.
This geometry Klein called the Hyperbolic geometry and the spherical and
Euclidean geometries he called Elliptic and Parabolic respectively, making the
distinction between them depend upon whether the right line has two real,
imaginary, or coincident points at infinity. He called the measure of distance
of the Hyperbolic and Elliptic geometries, the general metrical determination,
and that of the Parabolic, the special metrical determination. He remarked
that the two may coincide at a point or in the neighborhood of a point, but that
at points at a distance from the point of contact, the general metrical determina-
tion is greater or less than the special, according as to whether the fundamental
conic is imaginary or real. He designated as measure of curvature the gi-eatness
of the respective gain and loss, and found that it is the same at every point, and
that it is equal to — 1/4 C'.
HISTORICAL DEVELOPMENT OF PSEUDOSPHERICAL SURFACES. 45
He regarded all the points and lines on the plane as the projections of lines
and planes iu space and the absolute as the section of a cone whose vertex lies
at a determined point in space and which passes through the circle of infinity,
and was able to prove that projective geometry can be completely developed,
although absolutely free from the question of metrical determination. He thus
showed that the hyperbolic geometry, since it has a real value for C is the geom-
etry of surfaces with constant negative curvatui-e and that the non-Enclidian
geometry, the pseudospherical geometry and the hyperbolic geometry are essen-
tially one and the same.
II
THE SURFACE OF CENTERS AND THE TRANSFORMATION
OF PSEUDOSPHERICAL SURFACES.
1. The theorem that oo' new pseudospherical surfaces may be derived from
oue that is known and the geometrical method for the determination of the new
surfaces were derived by Biauchi**' for a simple case only in 1879. In 1881 the
theorem was developed analytically so as to apply to a more general case by
Backlund*" and was geometrically interpreted for this general case by Bianchi""
in 1887.
In its generalized form the theorem may be stated as follows : if on a surface
of constant negative curvature — 1/ H" a system of lines be chosen whose prin
cipal normals at every point make a constant angle (jr/2 — {A)J
where cj) and -ff'are functions of w and v of such a nature that they define the
principal radii of curvature R^ , i?., of the surface by means of the equations,
^j = ^(A"), R., = 4>{Ii)-K(ji'{K).
HISTORICAL DEVELOPMENT OF PSEUDOSPHERIOAL SURFACES. 49
The expression for the linear element of the Tf-surface itself, when exjjressed in
this notation, then becomes
*.. (iim,,„-+( i'('^-\7/sf^''^ )'d^.
P)
K )- ^\ ^\K)
Beltrami -" in a series of articles on the application of analysis to geometry,
published in 1865, proved both Weingarten's theorem and its converse and for
the latter found that a ruled helicoidal surface forms a case of exception, for
although this surface is applicable upon a surface of rotation, it cannot be the
nappe of a surface of centers. He showed at this time that the curves on the
nappe of a surface of centers which are enveloped by the normals to the involute
surface are geodesic lines, he therefore remarked that "if the geodesic lines of an
evolute surface become right lines the tangents to them at every point instead
of filling all space reduce to a system of straight lines with a single parameter
and are not sufficient to generate an orthogonal surface " ; he discovered rather
that, then, the geodesies themselves can generate a ruled surface which if it is
applicable upon a surface of rotation is applicable upon the minimal surface of
rotation, the cateuoid, and is parallel to a series of pseudospherical surfaces
instead of being a nappe of their evolute surface.
Diui-' also investigated this case of exception to the converse of Weingarten's
theorem in his paper on helicoidal surfaces, in the same year. He found that
the ruled helicoid that is applicable upon a eatenoid is a screw-surface generated
by a right line that moves along a helix lying on a cylinder, making a right
angle with the helix at every point and a constant angle with the cylinder, and
that it may be regarded as the locus of the normals of another helicoidal surface
upon which these same helices lie.
In the same treatise on the application of Analysis to Geometry-" Beltrami
made known several very important theorems concerning the surface of centers.
He denoted a surface whose principal radii of curvature are functionally related
by aS', its lines of curvature by m = a constant and y = a constant, its principal
radii of curvature and the two nappes of its evolute surface corresponding to
those lines of curvature respectively by R^, R^ and S^, S^.
First, he demonstrated the general theorem that if two systems of curves, one
of which is composed of geodesic lines, be conjugate to each other and if tan-
gents be drawn to two of the geodesic lines that lie infinitely near each other at
points a, and «., where they meet a curve of the other system, these tangents will
meet at a point which is the center of geodesic curvature of a curve which passes
through the point «, , and is orthogonal to all the geodesic curves of the first
system. Applying this proposition to the case of the nappes, ;S'i , S^ of the evo-
lute surface of a IF-surface he obtained the results first, that, since the normals
to the surface *S' taken along its lines of curvature u = a. constant touch the first
50 E. M. CODDINGTON.
nappe of the surface of centers, S^ , along a family of geodesic lines, which are the
evolutes of these lines of curvature and which may also be denoted by m = a con-
stant, and the normals to the surface taken along the lines of curvature v = a con-
stant, are tangent to this same nappe along curves which, as Kummer has shown,*
are conjugate to the geodesic lines u = a constant, S^ is the locus of the centers of
geodesic curvature of the orthogonal trajectories of the geodesic lines ?/. = a con-
stant on S^ and, conversely, the centers of geodesic curvature of the orthogonal
trajectories of the geodesic lines corresponding to u = a constant on S^ lie on
/Sj ; second, that the difference between the principal radii of curvature of the
involute surface S at any point J^ is equal to the radius of geodesic curvature
at a corresponding point of the orthogonal trajectory of the curves on either
nappe which are the evolutes of the lines of curvature of the surface S, thirdly,
that, when k^ denotes the arc length of a curve in the nappe S\ which goes
through any point ^j corresponding to p and which is the evolute of a line of
curvature ?« = a constant in the surface S and when p denotes the radius of
geodesic curvature for the point p of a curve orthogonal to Wj and going through
the point p , the principal radii of curvature 7^ and i?., of the surface S a.t P
are given by the equations
Beltrami's^'' direct contribution to the subject of pseudospherical sui-faces at
this time consisted In the determination of their evolute and involute surfaces.
He first found the equation of relation connecting the principal radii of curva-
ture i?j and i?2 °^ ^'^y TF-surface defined by the equation
and the principal radii of curvature Ii[ , i?^ of the surface of rotation on which is
developable one of the nappes of its evolute surface. He wrote the equations for
Ii[ and i?.', in the usual form for the principal radii of curvature of the surface
of rotation.
aI-(^)
jX , — "~~ 70 > Jlij
1- fJ2 y -2- r
where r^ = the radius of a parallel circle and ?/, = the arc of a meridian curve.
Substituting in these equations the expressions for i\ and drjdu^ ,
= e
/—""i— clr, r, (Fr, 1 du,
' J?/, ?(, - <|) ( ?,
where A;- = = for the three surfaces respectively.
These results led him to announce the theorem that " a sui'face complementary
to a surface of constant negative curvature with respect to a system of geodesic
lines which go out from a point on the surface is developable upon a rotation
surface which has for its axis the asymptote and for its meridian curve, a cur-
tailed tractrix, the ordinary one or an elongated one, according as the point of
intersection is finite and real, at infinity, imaginary."' " The first named
curve," he said, "is none other than the orthogonal projection of the tractrix
upon a plane which goes through the asymptote. On the other hand, the last
named curve has the tractrix for its orthogonal projection."
He further observed that the deformed parallels of the surface of rotation upon
which the complementary surface is applicable correspond to the deformed parallels
of the surface of rotation into which the original surface is developable. He
proceeded to find the equations for surfaces other than surfaces of rotation that
are complementary to a pseudosphere with respect to a family of geodesic lines
of each of the three kinds, and having found these equations he showed that the
corresponding surfaces are applicable upon one of the three kinds of surfaces of
rotation. Bianchi"' also extended the application of this theorem to helicoidal
surfaces, and found that there are also three kinds of helicoidal surfaces com-
plementary to a pseudospherical helicoid corresponding to the three kinds of geo-
desic lines with reference to which they may be developed.
Since the surface complementary to a pseudospherical surface with respect to
a system of geodesic lines going out from a point at infinity is developable upon
a pseudosphere and has the same curvature as the original surface, and since there
are oo' systems of this sort on a surface of constant negative curvature, Bianchi
remarked that from a pseudospherical surface >S'| an infinite number of new
surfaces S., of the same curvature may be derived, and that from each surface
Sr, an infinite number of new surfaces tS^ , also with the same curvature, may be
obtained in the same way as S^ is obtained from >S'., , provided that a fanaily of
geodesic lines on S^ are known, and so on.
6. From Bianchi's surface that is complementary to a pseudospherical surface
with reference to a family of geodesic lines going out from a point at infinity,
Kuen, in 1879, derived by the repetition of Bianchi's operation the equations for
a new pseudospherical surface which he classified as an Ennejier surface. The
paper in which Kueu*'^ announced these results was referred to on page 26.
In the same year Lie^'"^ developed Bianchi's theorem further. In a paper
66 E. JI. CODDINGTON.
publishecl in the Arcbiv for Mathematik nnd Naturvideuskab he
introduced a method for finding by means of a quadrature alone the geodesic
lines of the surface of centers of a TF-surface, and especially for the case when
the surface of centers is composed of pseudospheres. " This problem for deter-
mining the geodesic lines," he observed, "is equivalent to determining the lines
of curvature on the TF-surface."
He supposed the TF-surface to be referred to a system of curvilinear coordi-
nates {x, y) and expressed one of its principal radii of curvature R^ at a point
and the coordinates x^ , y^ , 2^ of a point on the corresponding nappe S^ of its evo-
lute surface as functions of these parameters. He wrote the expression for the
linear element on the nappe S^ referred to a family of geodesic lines and their
orthogonal trajectories as parameters and of curvature — l/«', in the usual form
From this he derived the equation for the geodesic lines
dv = e-^''« V{ds\- dE\ ) = e-^'"- Vdx] -f dy\ + dz\ - dR\
and observed that the quantity under the sign of the radical is of the foi-m
{X{x,y)dx + Y{x,y)dy]\
where JTand l^are functions of x and y only, so that he could at once obtain
the integral of the equation containing an arbitrary constant. Therefore, if he
had given a surface S^ with curvature — l/a", he could bring its linear element
in 00 ' ways into the form
ds\ = dR\ + e-''''"dv- *
referred to a family of geodesic lines going out from a point at infinity and their
orthogonal trajectories, and considering this surface as the first nappe of a sur-
face of centers he could derive an involute surface corresponding to one of those
infinite systems of geodesic lines. From this involute surface he could obtain
a second nappe S^ of a surface of centers and by the method just given deter-
mine on it a family of geodesic lines and write its linear element in the form
ds'i = dR^ + e--''-"du°.
A repetition with respect to S„ of the operations performed on S^ would then
enable him to obtain a new set of surfaces S^ and by the successive application
of this same process he could derive oo * surfaces all with the same constant
negative curvature — 1/a". In actual practice he remarked "it is possible to
go directly from one nappe to a second without stopping to obtain the involute
surface."
It may here be remarked that several years later in 1888 Weingarten '"'
developed another method for finding the lines of curvature on a TT'^surface and
* Page 12.
HISTORICAL DEVELOPMENT OF PSEUDOSPHERICAL SURFACES. 57
consequently the corresponding geodesic lines on its surface of centers, wliicli,
according to Darboux,'^'' ' "''\ is "more precise but less direct "' than that of
Lie.
After having developed this method for determining the geodesic lines of a
pseudospherical surface Lie next called attention to a method for transforming
one surface into another that had been discovered by Bonnet many years before.
Bonnet^'''"' had shown that every surface of constant mean curvature is
applicable upon an infinite number of surfaces of the same sort and that such
a surface is parallel to a surface of constant total curvature and obtainable
from it by dilatation. Lie suggested therefore that if a parallel surface be
derived from a pseudospherical surface and transformed into an infinity of new
surfaces with constant mean curvature and each of these in its turn be trans-
formed back into a pseudospherical surface, the result will be the same as if
Bianchi's operation had been performed upon the original surface of constant
negative curvature.
He showed moreover that the asymptotic curves of a surface of constant curva-
ture may be found by a simple integration and that they correspond to the
minimal lines of a parallel surface, a theorem which furnished the means of
obtaining the equation of transformation as it is given by Darboux.*
Lie used for the linear element of the surface referred to its lines of curvature
?/ = a constant and v = a. constant the expression due to Weingarten •" f
and for the asymptotic lines of the surface u^ = a constant, and c, = a constant
the corresponding expression
This last equation he saw is integrable if
m-
{ inconstant).
4>"
The general integral of this equation is
4P- = AK~ + LA-,
and the total curvature of the corresponding surface is constant for
■Ri^i^ ^^ -^-4" (i?i, if2= principal radii of carvatnre).
while a singular integral is
*Daebodx,>59 §775.
t Page 49.
58 E. M. COPDINGTOX.
and the mean curvature of the corresponding surface is constant for
^= j^ = ( i^i, i?j = principal radii of curvature).
He did not prove his theorem in detail nor give the equations of transformation
deduced from Bonnet's theorem, but Darboux, i^^, ?r7s j^ jjjg celebrated work con-
cerning surfaces, gives a simple proof for the correspondence between the asymp-
totic lines on the surface with constant curvature and the minimal lines on the
surface with constant mean curvature and, then, denoting the linear element on
the first surface referred to its asymptotic lines as parameters by
ds- = da- + 2 cos codad^ + d^'
and the linear element of a parallel minimal surface referred to its minimal lines
as parameters by
ds- = ie'^dadlS,
where 2&) is the angle between the asymptotic lines and also the angle between the
minimal lines, he pointed out that, when either surface is transformed into a new
surface of the same kind, the equations of transformation will be
n{a,/3) = co(^^^,a/3\
where 2fl is the angle between the asymptotic lines of the new surface with con-
stant curvature or the angle between the minimal lines of the new surface with
constant mean curvature and a is a constant.
The next year Lie''' raised the question whether the surfaces obtainable from
one that is known by Biauchi's method of transformation are all distinct from
each other or whether a finite number. of them are coincident, or as he expressed
it, "whether those surfaces of constant curvature 1/a", which are derived by
the infinitely repeated successive application of Bianchi's operation from one
that is given, must satisfy still other differential equations beside the equation
rt — s'- 1
He answered this question in the negative and his method of proving his answer
correct applies to surfaces of constant positive curvature as well as to those of
constant negative curvature.
He began his demonstration by writing down the equations which represent
known characteristic properties of the nappes of an evolute surface with con-
stant negative curvature
HISTORICAL DEVELOPMENT OF PSEUDOSPHERICAL SURFACES. 59
(x - .r, y + {!/- !/, )- + (s - s, )- = cr,
Pii-'^ - ^\) + ?i(y -yi) - ( ' - -i) = ,
l\n + q^q + 1 = ,
where (a;, y, :i,p^ q) and (.Cj, y^, z^, p^, yj determine an element on the first
and second nappes respectively.
The first equation shows that the distance between corresponding points on
the nappes is constant, the last three, that tangent planes to the nappes at cor-
responding points meet at right angles along a common tangential line. In this
way he showed, as Biicklund ■" remarked, that the surface on whose tangent
planes lie the circles with constant radius of a Ribaucour cyclic system and the
family of surfaces noi-mal to the circles are identical respectively with Bianchi's
initial sui-face and the infinite number of its complementary surfaces, for these
equations express analytically the fact that a system of surfaces,
are orthogonal at the point (a;, //, s) to a family of circles with constant radius
a and with their centers lying on the lines of curvature of a surface
»=/(-''■' y)-
Lie called his initial system of equations the equations of transformation,
and since he had four of them from which to eliminate the five variables, he saw
that to every element [x, y, z,j), q) there corresponds an infinity of elements
(cCj, y, , «j, Pj, 5'j), so that to the co- elements that go to make up the original
surface there corresponds oo' new elements, and he proved that these oo' new
elements can form oo' surfaces when the curvature is constant.* By applying
his equations for transformation he was thus able to obtain oo' surfaces <^, from
one surface i^and from these derived surfaces (j)^, oo" new surfaces F.,, among
which may be the first surface F. By repeating successively this operation he
finally obtained oc-'" surfaces F and 00-'" + ' surfaces (j>, for the surfaces of the
one class are finitely distinct from those of the other, but he had still to decide
whether he could derive all the pseudospherical surfaces in this way, or only a
limited number of them.
He considered two surfaces, i^and i^,, which differ so little that the one may
be deformed into the other by an infinitesimal transformation. By carefully
working out the equations for this infinitesimal transformation he found three
different equations for Sp and Bq , the increments of p and q , for determining
the way in which an element x, y, z, p, q passes into its next adjacent position.
He then assumed that this element could not go over into all the new elements
* Cf . page 65.
60 E. M. CODDINGTON.
but only into a certain number of them which form a locus defined by the equation
/(;o, 2/, z,2), 5) = 0.
He saw that this locus must be deformed into itself by the same operations
which transform the elements iufinitesimally and that, therefore, it must satisfy
three equations of condition, one corresponding to each of the three different
pairs of value of Sjj and hq . But from these same equations of condition he
found that the partial derivatives of the first order of f with respect to each of
the five variables vanish indejiendently, that consequently the locus f cannot
exist, but that each element passes over in aU the new elements and the given
surface is deformed by the equations of ti-ansformation into oc' new surfaces.
In like manner he found that the given surface can satisfy no partial differen-
tial equation of the second or third order and accordingly may be transformed
into 00^ or oo'' new surfaces, but he could not arrive at any general result by
this method. He""' next turned to the consideration of a strip on the given
surface formed by an aggregation of successive elements and, thex^efore, trans-
formable into 00 ' new strips. He proved by the actual application of the
equations for a Bianchi transformation that, if the curve C, formed by the
points of all the surface elements along a strij) is an asymptotic curve it may be
deformed into oo' new asymptotic curves JZ, that the arc length of each new
curve A' is equal to the corresponding arc length of C and that the curvature
l/i?j of each new curve is related to the curvature 1/i? of C by the equation
a a .
-^=-^-2sm..
where p is the angle that a line joining a point on the one curve to a corre-
sponding point on the other curve makes with the tangent to either curve at the
point where the curve is met by the line. He derived oo" new asymptotic
curves C, from the curves A'; by a third repetition of the oj)eration he obtained
00* new curves A', and so on, so that the problem as in the case of surfaces
resolves itself into the question, is there any limit to the number of asymptotic
curves that are thus derived? Reasoning in the same way as for the infinitesi-
mal transformation of surfaces, he found that the number of asymptotic curves
that can be derived from one that is known will be reduced, only, if these derived
curves can satisfy an ordinary differential equation. Denoting a/Ji by v and
a/H^ by i\ he wrote the equation connecting these values in the form
V = i\ — 2 sin V
and the equation for v in the form
dv
c ^- = — t), -f sm V .
as '
HISTORICAL DEVELOPMENT OF FSEUDOSPHERICAL SURFACES. 61
He then proved that such an equation as
cannot exist, so that any asymptotic curve corresponding to u = a constant can
be transformed into at least oo' new asymptotic curves. He proved that there
is no relation between v and its derivatives of the second or third order with
respect to s nor, indeed, between v and its derivatives of any order for on
account of the form of the equations for the increment of i\ Sv and Sv" he could
write down by analogy the equation for Bv" and then show that the one for
Bv" + ' is exactly similar. He thus showed that there is no limit to the number
of asymptotic lines that can be obtained from a given one by the equations of
transformation.
He then turned back to the case of the surfaces and, by means of his new
results, increased the number of surfaces that can be derived from a known one
which passes through two intersecting asymptotic curves from oo^ to oo* thus
establishing his theorem.*
Lie proved that there is not only a correspondence between the asymptotic
lines on a transformed surface with those on the initial surface but also one
between their lines of curvature, for since, according to Dini's discovery, the
asymptotic lines of a surface divide it into lozenges, a net-work of lozenges on
one surface S is transformable into a net-work of lozenges on each of the derived
surfaces, and the lines of curvature which are their diagonals pass over into
lines of curvature.
During these same years from 1879 to 1882 while Bianchi and Lie were
making their important investigations on the method of obtaining new surfaces,
of constant curvature from a given one, Biiekliind '"■ " was publishing the results
of his studies on the transformation of surfaces in successive volumes of the
Mathematische Annalen and the discoveries of Bianchi and Lie were made
just at a time when Biicklund could use them as examples to illustrate his
theorems.
Among other propositions Biicklund'" considered the question whether two
surfaces may be transformed into each other when the relation between them is
of such a nature that it is defined by four arbitrary partial differential equations
of the first order.
He denoted the two surfaces by
z =4>{x, y) and z' =f(x', y'),
and using ^>, ^, ?■, s, ^ to denote the partial derivatives of the first and .second
order of z with respect to x and y as is customary, andp', q' , r', s' , t' to denote
* Bianchi, ■» ? 247.
62
E. M. CODDINGTON.
the jmrtial derivatives of the first and second order of z with respect to x and
y' he wrote the four partial differential equations in the form
F,{x,
y.
8,iJ,
q,x
.y\
z
^P'
l') =
0.
F,{
) =
0.
F,{
) =
0.
K{
) =
0.
He then proceeded to find under what condition the surface whose equation is
z = 4,{x, y)
may be transformed into the surface whose equation is
z =f(x', y')
by means of these equations, he first substituted in the equations i^^ = and
i^j = the values of z,p,q expressed as functions of x and y . He then solved
the resulting equations for x and y expressing them in terms of the accented
variables only. By means of these results he could, by making the proper
substitutions, reduce the last two equations
i^3 = and F^ =
to equations containing a;', y', s', p ^ q only, in which case he denoted them by
i^; = and i^; = .
He could then obtain the function z by means of these equations provided that
they are compatible. The equation of condition which must be satisfied by z
when the two equations
are compatible may be obtained by first taking the total derivatives of each of
these equations, which are also equal to zero, then solving the resulting equations
for dp and dq so that
dF'. dF:
dF'^ ,dF'. dF'.
3 3
dx' ^i dz' ' dq'
dq ' dp
dp' =
dx' +
dF',, dF\
i.
dF'^ ,dF'^ dF'^
-~ +r> --1- , ^--^
dx '■ cz cq
dq ' dp'
dF'^ dF'^l
dq =
dF'. ,dF'^ dF'^
3 1 ' 3 i
dx ^^ dz ' dp'
dx' +
dF', dF\
dF\ ,dF'^ dF'^
4 4
dq ' dp'
dx' '^^ dz' ' dp'
dF', ,dF', dF',
cy ^ cz dq
dF\ ,dF\ dF',
cy ^ cz cq
dy'
+ 9^,
dz' ' dj)'
dF', ,dF', dF\
dy ^ cz cp
dy'
dy'
and finally setting the coefficient of dy' in the first of these last equations equal
HISTORICAL DEVELOPMENT OF PSEUDOSPHERICAL SURFACES. 63
to the coefficient of dx' in the second equation from which results the equation
{dF;AdF^_/dF^\dF: (dF^\ BF: _(dF\\ dF[ _
\ dx ) dp' \ dx ) dp' + V ~(iy' ) ^q \ H ) ^q' ~
where
/dF'.\ dF'. ,dF'. ,dF'. ,dF'.
m-
dz' ^ dp' ^^ Bq'
{i = 3, 4).
dF'. dF'. ,cF'. dF'.
-dy' +^-d7 +'^dp^+'^
This equation he represented by the bracket \_F'^F'^'\f,,,p.= and when instead
of F'^ and F'^ he introduced their equivalent values in terms of the unaccented
variable the eciuation became
IF'.F',] = [i^3^J=.v + '2^[»^i^J +^'[2/^J +^^^[^3^]
or finally
dF.^^ ^ VdF,dF. dF.dF.l^ , „
[F:F,]=iU)[F,F,],^,^,+ {i2)[F,F,],,,,.+ (2S)[F,FJ,,^,
+ (12) [F^F,],,^, + (13) [F,F.,],^,,. + (14) [F,F^],^,„ = 0,
where
, , /dF \ (dF \ (dF \ (dF \
He had. therefore, three equations
containing the accented variables only, which will determine a surface vJ = /"(.r', y')
and only one surface, provided that these equations are in involution, a condition
which he represented in the usual manner by the equations
He moreover showed that the function z =/'(.t', j/' ) will satisfy two partial
differential equations of the third order obtained by eliminating cc', ?/', «', p ^ q
from the four equations of transformation and from the equations of condition
\_F'^F'^'\ = 0. He remarked that an excejition to this theorem occurs when
z' does not appear in the equation
[F',F',] =
and that then x^ surfaces 2' =f(^x',y') will correspond to one surface 2 = ^(x,?/),
*z'x'p' written after the limcket signifies tliat Fi is differentiated witb respect to the
accented variables only.
64 E. M. CODDINGTON.
for, in that case, instead of two partial differential equations of the third order
for z there will he one single partial differential equation of the second order,
and if an integral of this equation he substituted for s in the equation of trans-
formation, the quantities x\ y' ip' ■, q can be expressed in terms of x, y, z so that
the function .-' will be determined by an equation of the form
dz = A{x, y, z')dx + B(x, y, z')dy.
The integral of this last equation will contain an arbitrary constant which proves
the theorem that there are oo^ sui'faces z' =y'(.c', y' ) corresponding to one sur-
face z = 4>{x, y).
BKcklund"'' saw that a surface transformation of this nature occurs in Bianchi's
problem for deriving a surface complementary to a known surface of constant
negative curvature. He considered the two surfaces defined by
s = c^(x,2/) z'=f{x',y')
as the two nappes of the evolute surface whose radii of curvature are connected
by the relation
The two relations existing between the two nappes, that the distance between
corresponding points is a constant a and that their tangent planes at correspond-
ing points must meet at a right angle along the common tangent, gave him his
four equations of transformation
F, = p{x' - X) + q(y'-y)-(z'-z) =
F, =p' {x -x) + q {y' —y)—{z'-z) =
F.^=l-{-pj/ +qq' =
F^ = ix-x'y + {y-y'y + {z-z'f-(r=0.
and his equation of condition took the form
(rt - s"-) + «-(l + 2r + q-y- = ,
since the expressions [F^F^],^,^„ [F,F^],^,^„ [F,F,],^,^„ [F,FJ,,^, all
become equal to zero. He saw from this equation and from a similar one for z,
since the equations of transformation are symmetrical with respect to the
accented and unaccented variables, that both surfaces are of constant negative
curvature — l/a' and that since z' does not appear in the equation of condition
that there correspond an infinity of surfaces z' =J\x' , y' ) to evei-y surface
z = (^(x, y).
In 1884 BKcklund ^' wrote an important paper that deals exclusively with
pseudospherical surfaces. In this paper, published in the Lund's Uui-
HISTORICAL DEVELOPMENT OF PSEUDOSPnERICAL SURFACES. 65
versifcets Arsskrift and entitled "Concerning- Surfaces with Constant Nega-
tive Curvature," lie first reviewed the contributions made to the theory of the
transformation of the pseudosjjherical surfaces by Bianchi, Ribaucour and Lie,
pointing out the close connection between the theories of Bianchi and those of
Ribaucour, he then extended Bianchi's theorem to fit a more general case, namely,
when the given surface and the derived surface are not the nappes of an evolute
surface but are so related to each other that planes tangent to them at corre-
sponding points cut each other at a constant angle, but not at right angles, and
the distance between two corresponding points is constant. He expressed this
condition by leaving the first three equations of transformation unaltered and
writino; i^, = in the form
^4 = 1+ pp' + n - ^^{ v 1 + f + r){ 1' 1 + p" + ?") =
where K is the cosine of the angle formed by the two tangent planes and is a
constant.
He then found that the equation for z becomes
rt -s-= - -— r- (1 + F + r)''
and that a like one exists for z' , so that in the general case also both surfaces
have constant negative curvature. By putting for (1— A'')/(r a constant
Ijm- and letting a and TT vary, he obtained an infinity of equations of transfor-
mation for surfaces only whose curvature is — \jm- and in particular those for
Bianchi's complementary transformation when Il = and m = a.
He made a complete study of this general method of transformation. First
he remarked that the set of equations
f'(x)-y!r(x)
«=/(x),y = c^(:«), I> = ir(x), q= ' "^ i'{x)
determine a curve on the surface together with the direction of the tangent plane
to the surface along that curve for successive values of x, that is, they determine
a strip of the surface. Then recalling Cauchy's theories he observed that if .r, y
and a are the coordinates of an arbitrary point in the strip and cb^, y^, z^ the coordi-
nates of its initial point, a surface passing through this strip and satisfying a
known differential equation may be defined by a convergent Taylor's series in
terms of {x — x^, {i/ — y^) where the singular points of the surface are not con-
sidered, thus
« - '^0 =Po{^' - ^o) + loiy - 2/o) +H'-o(a' - ^o) '+ 2s„(a^ - Xo)(y - y,) + t,{y - y.f}
+ 3j {u,{x- xj -f 3r„(x - o:J{y- y,) + 3«.,(.-.3 - .r„)(.y - y,f + o>^X>J - %Ti
+ ••••
66
E. M. CODDIXGTOX.
For the surfaces under discussion he obtained the values of the coefficients r, s,
etc., from the equations
dp = nix + sdi/, dq = sdx + tdy ,
SO that t is determined by the equation
dpdx + dqdy
In general only one surface can be found passing through the strip, but when
the value of t is indeterminate, that is, when
dpdx + dqdy = and dq- 2(1+ P' + q")'dx- = ,
Backhmd saw that there are an infinity of surfaces having contact of the first
order along the lines defined by those equations and that these lines are the
characteristic curves of the integral surface. Since, the first of these equations
shows that each curve may have for its plane of osculation at every point the
tangent plane to the surface on which it lies at that point and the second equa-
tion shows that the torsion of the curve is constant, Backlund thus proved that
the characteristic curves of surfaces of negative constant curvature are asymptotic
curves.
He, then, demonstrated geometrically that the guiding curve of every strip )•',
derived from a strip r on the original surface S, by means of the set of equations
of transformation satisfies a partial differential equation of the Kiccati type and
that, consequently, every strip ;•' corresponds to a solution of such an equation.
This final result may then be stated as follows : If the surface S is known, all
the surfaces S' may be derived from it, for every strip r on S passes over into
an infinity of strips »■', one on each surface S' , by means of the equations of
transformation, therefore each derived surface S' corresponds to the solution of
a Riccati equation and when one such surface is determined, all the others may
be found by quadrature alone, since that is the only operation required to obtain
all the solutions of a Riccati equation when one is known. Backlund has jjroved
geometrically that the asymptotic curves on the new surfaces *S" are the deformed
asymptotic curves of the original surface S and that they also satisfy an equation
of the Riccati form.
In 1883 and 1884, Biauchi ^'' *»• '"■ ''• »' published his investigations of Ribau-
cour's propositions concerning a system of surfaces which have a family of 00^
circles for their orthogonal ti-ajectories. Biicklund ^" had already observed that,
HISTORICAL DEVELOPMENT OF rSEUDOSPHERICAL SURFACES. 67
when the circles all have the same constant radius aiul lie in the tangent planes of
a known surface, this known surface and the surfaces orthogonal to the circles are
identical with a pseudosphere and the oo' surfaces derived from it by means of
a complementary transformation with respect to a family of geodesic lines on it
that go out from a point at infinity. Bianchi gave an exact proof of tlie iden-
tity of the two families of surfaces by establishing the theorem that a surface
orthogonal to a family of oo' circles, can be regarded as the nappe of the evolute
surface of a TF-surface, provided that the line of intersection of the plane of
every circle with the planes tangent to the ortiiogonal surface at its point of con-
tact with that circle envelopes geodesic lines on the surface and by showing that
those enveloped curves are geodesic lines when the radius of the circles is always
the same. Bianclii's construction of a cyclic system of surfaces is as follows :
Let /S'j* be a surface orthogonal to a family of circles. Let these circles lie
on the tangent planes of a second surface S.,, and let the points of tangency of
those planes with the surface S^ be the centers of the circles. Let u = a con-
stant and V = a constant denote the lines of curvature of the surface S.^ and let
6 be the angle that a radius of the circle, mn, drawn to meet an orthogonal
surface S^ at m makes at its center n with the line of curvature d = a constant
passing through that point. The radius of the circle mn being tangent to the
orthogonal surface S^ must lie in the tangent plane at m and is the line of inter-
section of the plane of the circle with the corresponding tangent plane of the
orthogonal surface S^. Let u' and v' be the lines of curvature of S^. Let (^
be the angle which this line of intersection makes with the tangent to the line
of curvature of ,S'| , u' = a constant, at the point of contact m. When the sur-
face S^ is regarded as known, each orthogonal su-rface S^ corresponds to a value
of 6. When an orthogonal surface ^S", is regarded as known, each circle is
determined by its radius and the value of (/> to which it corresponds.
Bianchi *^ denoted by = a constant, the curves on the orthogonal surface S^
which are the orthogonal trajectories of the curves on that surface that are
enveloped by the lines of intersection of the planes of the circles with the corre-
sponding tangent planes of the surface ^S'^. When the linear element of this
surface S^ referred to its lines of curvature as parameters assumes the form
dSl = Edu- + Gdv-,
he showed that must be a solution of the differential equation
d'^ d^ d^ 1 d\/'E d^ 1 dVG d^
dudv ~ du ' dv -^^ dv du •^'Q du dv'
which, if log Z is written in place of <&, reduces to an equation for Z identical
with that given by Kibaucour.
*Daeboux,>39 I 804; Bianchi,"" § 179.
68 E. M. CODDINGTON.
He also proved Eibaucour's statement, that it is necessary to be able to
integrate this equation in order to find all the C3"clical systems of which the sur-
face /S', forms a part, for he derived for li and (/>, the functions by which a
circle is determined, the following expressions
1 1 d^ 1 ccj)
-^„ = A. $,* cos = B — ^^^ , sin 4> = H — .-- ^^,
which can be found when the surface S^ and a value of (j) ai-e known.
From his equations for expressing the condition that the circles are orthogonal
to a surface S^, Bianchi was able to show that when the circles have all the
same constant radius i?, the surface /S', as well as the surface S,,, on whose
tangent planes the circles lie, wiU both have constant negative curvature
— l/M'. In that case he saw that
A, = a constant,
or that ^ = a constant ai-e geodesic parallel circles and that the curves enveloped
by the lines of intersection of the planes of the circles with the corresponding
tangent planes of the orthogonal surface /S*, and which are the orthogonal tra-
jectories of ^ = a constant will be geodesic lines. Moreover, he found that
the geodesic curvature, 1//3, of the curves = a constant is equal to 1/^ so
that when H is constant they are the deformed horicj'cles of the pseudosphere
on which the surface S^ of curvature — 1/i?- is applicable. The fact that the
curves on the surface aS, that are enveloped by the lines of intersection of the
planes of the circles with the corresponding tangent planes to the surface S^ are
geodesic lines was the only condition Beltrami required in order to prove that
the surfaces S^ and S.-^ form the nappes of an evolute surface, for these lines of
intersection, being tangent to the surface S^ along geodesic lines, may be regarded
as the rays of a normal congruence of which the surfaces S^ and S., are the
focal surfaces.
It is not necessary to give in detail the equations and theorems by means of
which Bianchi proved Eibaucour's propositions, that if a system of co ^ circles
are orthogonal to three surfaces they wiU be orthogonal to oo ^ surfaces and the
theorems relating to the triply orthogonal systems to which these surfaces
belong, but it is important to consider a proof given by Darboux*' in 1883 for
the establishment of the theorem regarding the existence of this triply ortho-
gonal system, for in that connection Darboux"'''''^'' developed for the first
time the now weU known set of equations for performing a complementary trans-
formation. He regarded as known the surface S„ of curvature — 1 on whose
tangent planes lie the circles of the system. He chose the lines of curvature
1 /d^\- 1 /c'i\^
*^'* = G(a;J +^(,-): BIANCHI -§§35, 86.
HISTORICAL DEVELOPMENT OF PSEUDOSPHERICAL SURFACES. 09
of this surface for its lines of reference and wrote the linear element in the form
ds' = COS" Q)du' + sin-
a formula which demonstrates the existence of the triply orthogonal system.
Bianchi"' obtained a like set of equations for representing a Biicklund trans-
formation. Employing the same expression for the linear element of the initial
surface referred to its lines of curvature as Darboux had used,
ds- = COS" o)du^ + sin^ ccdv',
and denoting by a- the complement of the angle between the tangent planes at
corresponding points of this surface and a derived surface, he first wrote these
equations in the form :
89 dco sin 6 cos to -\- sin a cos 6 sin to
du dv ~ H cos a- '
36 c(o cos 6 sin co -\- sin cr sin 6 cos w
dv du It cos cr
and, by using asymptotic lines on the initial surface for parameters instead of
lines of curvature, reduced them to the simpler form
d(6 — a) 1 -f sin o- .
-A- > = ^ sin (^ -f o)),
CMj M COS sni ( ^ — ft) ) ,
dU| It cos a ^ '
He saw that these equations are compatible if the curvature of the initial surface
is constant and negative, and that they form a Riccati equation for tan ^/2 such
as Biicklund had obtained previously.
Bianchi represented a Biicklund transformation by B^ and his own, or the
complementary transformation, when o- = 0, by jB,,. Later he denoted a Lie
transformation in which, retaining the previous notation.
0(w, v) =a)f rnf, \
* Page 58.
HISTORICAL DEVELOPMENT OF PSEUDOSPHERICAL SURFACES. 71
by L, and wrote
1 4- sin a 11 — sin cr
a = — and ~ = .
cos S| and S^ denote its two focal surfaces, that is, S^ and S^ denote the two focal
surfaces of a ray system which is characterized by the property, that the distance
between the limiting points on every ray is equal to a constant R and the dis-
tance between the points of intersection of every ray with the focal surfaces is
also a constant and equal to It cos cr, where (7r/2 — S\ are connected by the operations for a Biicklund
transformation so that the one surface S^ may be transformed into the other
surface *S', by this method, that the angle 6 has the same significance for >S'j as
(I) has for S.^ and that the linear element of