PURE MATHEMATICS 177 



observer, and have been ignored — except by Prof. Eddington — 

 in other discussions of relativity theory. PreHminary postulates 

 of the theory have physical interpretation, but a strict mathe- 

 matical structure is built up from them, and the question raised 

 by the relativity theory, in its recent generalised form, is as to 

 whether the postulates of Dr. Robb as interpreted are correct 

 expressions of phj^sical fact or only approximations. 



In a very valuable, though short appendix, the author gives 

 a more analytical discussion of the relation between the two 

 theories. Let us imagine that the relations of " before " and 

 " after " have an interpretation which might differ from the 

 optical one previously used by the author. On one of his 

 inertia lines, when measurement is introduced, it is found 

 that the formula — 



ds'- = df — dx' — dy' — dz' 



is valid, which in polars is 



ds' = — dr' — rW — f'sin'O d(j)' + dP 



A modified interval measure may now be introduced which 

 changes ds^ into dsi^ where 



d ^= d ' i — ^^ (^y 2^'^ (-^^X 

 \ r — 2m \ds} r^ \ds/ J 



and from this follows Schwarzschild's formula belonging to 

 the region round a single spherical body according to Einstein's 

 gravitation theory. The author compares this with the 

 brachistochrone problem, where the modified interval measure 

 is the time taken to traverse it. 



The author then shows quite simply that all the difficult 

 geometric systems involved are constructed easily from his 

 relations of before and after, by the use of a modified interval 

 measure. These relations appear, in fact, though the author 

 gives as yet no complete proof, to be in some sense true with 

 or without the presence of matter, but that the postulates of 

 the theory can only be interpreted strictly in the optical manner 

 when appreciable quantities of matter are present. The es- 

 sential thing is the four-dimensional manifold, and the geo- 

 metries become analytical developments which are suitable 

 for various special purposes. The author is clearly opposed 

 to the conception of " curvature of space," which, from this 

 standpoint, is not an implication involved in the use of any 

 such geometries. 



One very fundamental feature of the logical scheme de- 

 veloped by Dr. Robb is that the idea of " congruence" is an 

 essential and intrinsic part of the system, appearing in a natural 

 manner, and not as something brought in from outside to a 



