The reader is expected to have a general acquaintance with the less technical discussion of the theory given in \Title{Space, Time and Gravitation}, although there is not often occasion to make direct reference to it. But it is eminently desirable to have a general grasp of the revolution of thought associated with the theory of Relativity before approaching it along the narrow lines of strict mathematical deduction. In the former work we explained how the older conceptions of physics had become untenable, and traced the gradual ascent to the ideas which must supplant them. Here our task is to formulate mathematically this new conception of the world and to follow out the consequences to the fullest extent. The present widespread interest in the theory arose from the verification of certain minute deviations from Newtonian laws. To those who are still hesitating and reluctant to leave the old faith, these deviations will remain the chief centre of interest; but for those who have caught the spirit of the new ideas the observational predictions form only a minor part of the subject. It is claimed for the theory that it leads to an understanding of the world of physics clearer and more penetrating than that previously attained, and it has been my aim to develop the theory in a form which throws most light on the origin and significance of the great laws of physics. It is hoped that difficulties which are merely analytical have been minimised by giving rather fully the intermediate steps in all the proofs with abundant cross-references to the auxiliary formulae used. For those who do not read the book consecutively attention may be called to the following points in the notation. The summation convention (\PageRef{50}) is used. German letters always denote the product of the corresponding English letter by~$\sqrt{-g}$ (\PageRef{111}). $\Ham$~is the symbol for ``Hamiltonian differentiation'' introduced on \PageRef{139}. An asterisk is prefixed to symbols generalised so as to be independent of or covariant with the gauge (\PageRef{203}). \PageSep{vi}