PHYSICS 189 



geodesic world-line of a particle. If the particle is subject to 

 mechanical pressures and pulls, or to electro-magnetic forces, 

 the path in a space-frame no longer corresponds to the geodesic 

 world-line. If we accept the very prevalent belief that all 

 mechanical forces, " cohesions," " affinities," etc., are at 

 bottom electro-magnetic, we see that a great step forward in 

 the geometrising process would be effected if we could connect 

 the Maxwell equations of the electro-magnetic field with the 

 metrics of the world, as clearly as Einstein's equations of the 

 gravitational field are connected. This has been attempted by 

 Hermann Weyl of Zurich. His original papers are published in 

 the Sitzungberichte d. Preussische Akademie (191 8) and in the 

 Annale7t der Physik, Band 59, pp. 101-133 (1919)- A very full 

 account of his theory is contained also in his book Raum, Zeit, 

 Materie (Springer, Berlin). English sources, of as " popular " 

 type as is possible, are Eddington's Space, Time, and Gravi- 

 tation (C.U.P.) and the new edition of Cunningham's 

 Relativity and the Electron Theory (Longmans). 



The essence of Weyl's idea is this : The laying down of a 

 co-ordinate system does not involve measurement at bottom, 

 but only numbering according to an ordered and consistent 

 scheme ; but the statement that such and such a function of 

 co-ordinate differences rneasures a distance, an interval of time, 

 a separation between two events, obviously does, and implies 

 the use of some gauge (a definite stick, e.g. for length, and the 

 interval required for light in gravitation-free space to travel 

 along it for time, and a " blend " of these [a " clock-ruler "] 

 for separation). But how are different observers to know that 

 they are using the " same gauge " ? Only by direct comparison. 

 This implies transportation of a gauge, or a strict copy of it, 

 from one place in space-time to another for comparison with a 

 gauge used there. But will the path by which it is transported 

 have any effect on it ? Weyl asks us to face the possibility 

 that it does, to free ourselves from the restricting assumption 

 that the path has no effect, to accept as conceivable that, e.g. 

 a ruler on being carried from A to B by a certain path might 

 be perfectly congruent with a ruler stationed at B, and yet 

 might not be so if transported by another path ; or, to put it 

 still more strikingly, that a strict copy of a ruler at A might on 

 moving to B and returning to A by a different path not fit true to 

 its original. To give analytical expression to this idea we 

 assume that the fraction by which the gauge alters in a displace- 

 ment from a point-instant x^, x^, x^, x^ to a neighbouring one 

 Xi -\r ^^1, etc., is 



<|), hx^ -{• <^g hx^ + <^a ^^3 + 4>i ^^4 



where ^^, ^^, ^^, ^^ are four functions of the co-ordinates. 

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