270 



WILSON AND MOORE. 



Section p^^^ 



35. The indicatrix . , , 324 



36. Minimal surfaces 325 



37. The intersection of consecutive normals 326 



38. The fundamental dyadic <I> 329 



.39. The scalar invariants 332 



Chapter III. Special Developments in Surface Theory. 



40. The twisted curve surface and ruled surfaces 336 



41. Developable surfaces 340 



42. Development of a surface about a point 342 



43. Segre's characteristics 347 



44. Principal directions 350 



45. Asymptotic lines 354 



46. The Dupin indicatrix 357 



47. A second standard form for a surface 359 



48. Surfaces of revolution 362 



49. Note on a vectorial method of treating surfaces 364 



1. Introduction. There are several ways of generalizing the 

 ordinary differential theory of surfaces. The one most extensively 

 treated is that which deals with varieties of n—1 dimensions in a 

 Euclidean space of n dimensions. •'■ A second method is to investigate 

 properties of two-dimensional varieties in a space of four or indeed of 

 n dimensions.^ A third and more general extension of the theory 

 would be to study varieties of k dimensions in a space of n dimensions, 

 and under this head a very interesting species can arise ^ when n = 

 2 k— I. The recent contributions to this third have dealt with the 

 projective differential properties and thus have afforded only a partial 

 generalization of the general theory. 



We propose here to study the theory of two-dimensional varieties 

 in space of n dimensions and to exhibit the way in which the ordinary 

 theory arises through specialization. The generalization in this case 

 is not so immediately obvious as in the first case and perhaps throws 

 more light on the ordinary theory of surfaces than does that. 



1 See, for instance, ICilUng, Die Nicht-Euklidischen Raumformen; Bianchi, 

 Lezioni di Geometria Differenziale, Vol. I, Chaps. 11, 14; Shaw, Ainer. J. 

 Math., 35, No. 4, 395-406. 



2 K. Kommerell, Die Kriimmung der Zweidimensionalen Gebilde, in ebenen 

 Raum von vier Dimensionen, Dissertation, Tiibingen, 1897, 53 pp.; and 

 Riemannsche Flachen in ebenen Raum von vier Dimensionen, Math. Ann., 60, 

 in which the dissertation is also summarized; E. E. Levi, Saggio sulla Teoria 

 delle Superficie a due Dimensioni immersi in un Iperspazio, Ann. R. Scu. 

 Norm., Pisa, 10, 99 pp.; C. L. E. Moore, Ann. Math. (2) 16, 89-96 (1915). 



3 C. Segre, Su una Classe di Superficie ecc, Att. Torino, 42 (1907), and 

 Rend. Circ. Mat., 30, 87-121 (1910); and further developments by Bompiani 

 and Terracini. 



