MATHEMATICS 



of the ellipse, just as the total energy is an invariant 

 for a set of bodies in motion. 



The Principle of Relativity is ultimately, on 

 its mathematical side, founded on the supposition 

 that the whole universe, considered as one system, 

 likewise has its appropriate invariants, which of 

 course cannot be completely unrestricted in nature, 

 but are subject to many obvious assumptions 

 suggested by known phenomena, which rule out 

 the majority of assumptions which would otherwise 

 be somewhat bewildering. In fact, in their choice, 

 the procedure was to take the formulas which 

 constituted a shorthand summary of existing know- 

 ledge, and to postulate invariants which would 

 not violate these and yet would leave a loophole, 

 by greater generality, to bring in gravitational 

 phenomena. The result was that Einstein found 

 only one possibility with, as I said, the pure 

 analysis very largely constructed long ago and 

 awaiting application. This possibility, when de- 

 veloped, constituted his theory. I hope that at 

 least the nature of the mathematical side of the 

 theory has been made somewhat clearer by these 

 brief remarks. 



If we turn now to another branch of my sub- 

 ject in which significant progress is at present being 

 made rapidly, we cannot perhaps do better than 

 devote a little attention to what are called ' asymp- 

 totic expansions.' They are the most frequent 



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