A selected list of original papers on the subject is given in the Bibliography at the end, and many of these are sources (either directly or at second-hand) of the developments here set forth. To fit these into a continuous chain of deduction has involved considerable modifications from their original form, so that it has not generally been found practicable to indicate the sources of the separate sections. A frequent cause of deviation in treatment is the fact that in the view of most contemporary writers the Principle of Stationary Action is the final governing law of the world; for reasons explained in the text I am unwilling to accord it so exalted a position. After the original papers of Einstein, and those of de~Sitter from which I first acquired an interest in the theory, I am most indebted to Weyl's \Title{Raum, Zeit, Materie}. Weyl's influence will be especially traced in \SecRefs{49}, \SecNum{58}, \SecNum{59}, \SecNum{61}, \SecNum{63}, as well as in the sections referring to his own theory. I am under great obligations to the officers and staff of the University Press for their help and care in the intricate printing. \Signature{A. S. E.}{10 \emph{August} 1922.} \PageSep{vii} \TableofContents \iffalse%**** CONTENTS PAGE INTRODUCTION 1 CHAPTER I ELEMENTARY PRINCIPLES SECTION 1. Indeterminateness of the space-time frame 2. The fundamental quadratic form 3. Measurement of intervals 4. Rectangular coordinates and time 5. The Lorentz transformation 6. The velocity of light 7. Timelike and spacelike intervals 8. Immediate consciousness of time 9. The ``3 + 1 dimensional'' world 10. The FitzC4erald contraction 11. Simultaneity at different places 12. Momentum and Mass 13. Energy 14. Density and temperature 15. General transformations of coordinates 16. Fields of force 17. The Principle of Equivalence 18. Retrospect CHAPTER II THE TENSOR CALCULUS 19. Contravariant and covariant vectors 20. The mathematical notion of a vector 21. The physical notion of a vector 22. The summation convention 23. Tensors 24. Inner multiplication and contraction 25. The fundamental tensors 26. Associated tensors 27. Christoffel's 3-index symbols 28. Equations of a geodesic 29. Covariant derivative of a vector 30. Covariant derivative of a tensor 31. Alternative discussion of the covariant derivative 32. Surface-elements and Stokes's theorem 33. Significance of covariant differentiation 34. The Riemann-Christoffel tensor 35. Miscellaneous formulae \PageSep{viii} CONTENTS CHAPTER III THE LAW OF GRAVITATION SECTION PAGE 36. The condition for flat space-time. Natural coordinates 37. Einstein's law of gravitation 38. The gravitational field of an isolated particle 39. Planetary orbits 40. The advance of perihelion 41. The deflection of light 42. Displacement of the Fraunhofer lines 43. Isotropic coordinates 44. Problem of two bodies---Motion of the moon 45. Solution for a particle in a curved world