Woolurd — Generalized Itelativity and Gravitation. 433 



then, no matter how we transform this equation, both 

 sides will transform themselves alike, and the equations 

 will persist in the form 



T'« = ¥'i (14) 



Consequently, if we once succeed in expressing the laws 

 of nature as linear relations between tensors and vectors, 

 and between tensors and tensors, we shall have satisfied 

 the postulate of general relativity. 



The transformation-formulas for tensors are easily 

 derived from those for the coordinates; "these trans- 

 formation-formulas express the components of the trans- 

 formed tensor as homogeneous linear functions of the 

 components of the original tensor. Therefore, if for one 

 system of coordinates a certain tensor is zero, it is zero 

 for any system of coordinates. Consequently, if once 

 we have expressed the laws of nature in the form of linear 

 relations between tensors they will be invariant for all 

 transformations. ' ' Einstein, with the aid of the calculus 

 of tensors, assuming (11) to be invariant, and adopting 

 the fundamental formula (7), has thus succeeded in satis- 

 fying his original postulate. 



12. The invariant equations of motion turn out to be 

 the ordinary well-known Newtonian equations with cer- 

 tain small terms added to make them invariant. Under 

 ordinary conditions the two theories differ so little that 

 the new theory is of little use to experimental science, 

 the magnitude of the numerous indicated effects being 

 so minute ; however, the general relativity theory can 

 hardly be called a pure mathematical speculation, as only 

 physical lines of thought have been f ollowed in its devel- 

 opment. Possibly observable consequences are indicated 

 which are of great importance to physics, mathe- 

 matics, electromagnetism, molecular physics, astronomy, 

 mechanics, etc. 



13. The final equations cannot be given here on 

 account of the extreme complexity of the notation 

 necessary. Reference may be had to Dr. Fokker's 

 paper 14 and that of de Sitter 15 , in which all subjects dis- 

 cussed in the second part of the present article will be 

 found fully developed, with full references to the original 



14 Fokker, A Summary of Einstein and Grossmann 's Theory of Gravita- 

 tion. Phil. Mag:. (6), vol. 29, pp. 77-96, 1915. 



15 De Sitter, On Einstein 's Theory of Gravitation and its Astronomical 

 Consequences. Monthy Notices E, A. S., 76, No. 9, pp. 699-728, 1916. 



