Discussion of various treatments which have been given to the non-static line element for the universe

Introduction. - In several previous articles,(1,2,3) I have shown the possibility of deriving a non-static cosmological line element to agree with the relatively uniform distribution of matter observationally found in the universe, and of using this line element to account both for the annihilation of matter taking place throughout the universe, and for the red-shift in the light from the extra-galactic nebulae. After the publication of the first of these articles, I learned from a conversation with Professor H. P. Robertson that he had previously published(4) a different derivation of the same general form of line element; more recently Professor de Sitter and Sir Arthur Eddington have both been kind enough to call my attention to the previous use of a non-static line element by Lemaitre,(5) and still more recently I have discovered the early treatment of non-static line elements by Friedman.(6)