DIFFERENTIAL GEOMETRY OF TWO DIMENSIONAL 

 SURFACES IN HYPERSPACE. 



By Edwin B. Wilson and C. L. E. Moore. 



Received March 25, 1916. 



TABLE OF CONTENTS. 



Section Page 



1. Introduction 270 



2. Methods of attack 271 



3. Quadratic differential forms 272 



Chapter I. Ricci's Method. 



4. Two types of transformation 274 



5. Generalization of vector analysis 276 



6. Sets of elements 277 



7. Transformations of sets of elements 278 



8. An adjoined quadratic form 280 



9. Dual systems 283 



10. Composition of systems 284 



11. Mutually reciprocal n-tuples 285 



12. A standard form for systems . 287 



13. Orthogonal unit n-tuples 288 



14. Transformations of variables 289 



15. Solution for the second derivatives 292 



16. Covariant differentiation of a simple system 296 



17. Covariant differentiation of systems of higher order .... 297 



18. Contravariant differentiation 299 



19. Properties of covariant differentiation 300 



20. Relative covariant differentiation 301 



Chapter II. The General Theory of Surfaces. 



21. NormaHzation of element of arc 303 



22. Normal vectors 304 



23. Gauss-Codazzi relations 305 



24. Extension to n > 4 307 



25. The vector second form 308 



26. Canonical orthogonal curve systems 310 



27. Expressions of the second forms 311 



28. Moving rectangular axes 312 



29. Tangent plane and normal space 314 



30. Square of element of surface 317 



31. Geodesies 318 



32. Curvature; interpretation of a and -y 319 



33. Interpretation of P and |a 321 



34. The mean curvature 322 



