PHYSICS 187 



assumption was that the values of these g- " potentials " were 

 fundamentally determined by the gravitational properties of 

 the matter in the universe. Of course, if we change our frames 

 of reference, a mathematical transformation is involved which 

 alters the values of the ^-coefficients ; but that is natural, for 

 in the new frame we have a new (fictitious) gravitational field 

 imposed on the former one. The essential point is that, at events 

 not distantly removed from all matter, it is impossible to choose 

 co-ordinates so as to transform away all the differences between 

 the ^-coefficients and the simple (Lorentz) values. 



On a surface there are natural " tracks " for a particle 

 between two points, viz. the " geodesies," or curves for which 



\ViSn dx* + 2 ^1, dx^ dx^ + ^s,8 dx^) between the two points 



is " stationary " (maximum or minimum). Einstein took it 

 for granted that the track of the " event," consisting of a 

 moving particle, in space-time could be determined by a similar 



law. I a/( -S go.^ dxo. dxA between two given point-instants 



would be stationary for it. The actual orbit of the particle in 

 any observer's space would then be the projection on his space 

 of this " world-line," or " world-geodesic." The solution of 

 problems in orbits would then become a matter of Differential 

 Geometry of four dimensions, provided the value of the ^- 

 potentials were known. As a guide to the discovery of these, 

 Einstein submitted a certain set of differential equations, 

 known as his law of gravitation. We can briefly summarise 

 this by introducing certain symbols, as follows : 

 The determinant 



611 512 618 6 14 

 ^81 SzZ Sti §U 



Ssi Sat Sas Sst, 



Stl Sli St3 ^44 



is called g ; and the co-factor of any constituent g^,^ of the 

 determinant when divided by g is called g''''. 



[\/M, r] - i {^gxJBx^ + Bg^JBxx - Bg^JSx^) 



{X/A, y} - i; g^^lXfji, a]. 



Now Riemann had proved that in the geometry of any mani- 

 fold there were certain conditions that the manifold should be 

 " flat," or " homeloidal," i.e. that co-ordinate systems analogous 

 to Cartesian should be possible for it. They were the following 



differential equations : 



4 4 



b/bXfj, {\v, k) — b/bXv {Xyu., k} + H {\v, a} {afx, k} — ^ {\fi, a} {av, k ^ o{l) 



a = l = 1 



There are apparently 256 of these when we give k, X, ^, v the 



