president's address — SECTION A. 9 



It is interesting to notice that on 21st February, 1870, W. K. Clifford 

 read before the Cambridge Philosophical Society a paper on The Space- 

 theory of Matter. This paper is obviously inspired by Rieniann's 

 geonretrical work, and in it Clifford attributes all physical phenomena 

 to curvature, not of a four-dimensional continuum, but of the ordinary 

 three-dimensional space. He states : ■" I hold, in fact — 



(1) that small portions of space are in fact of a nature analogous 



to little hills on a surface which is on the average flat ; 



(2) that this property of being curved or distorted is continually 



being passed on from one portion of space to another after 

 the manner of a wave ; 



(3) that this variation of the curvature of space is what really 



happens in that phenomenon which we call the motion of 

 matter ; 



(4) that in the physical world nothing else takes place but this 



variation." 



These \'iews are not exact anticipations of those of Einstein, but they 

 are sufficiently like them to be worthy of notice. It is possible that 

 had Clifford applied them to the study of gravitation, instead of double 

 refraction, the recent development of Gravitation Theory might have 

 come from the University in which the Theory originated. 



To return to the main subject, Einstein first applies his Law of 

 Gravitation to the determination of the field of an isolated particle. 

 He uses polar space co-ordinates and chooses his units so that c = 1 . 

 He assumes that there will be s])ace symmetry about the particle and 

 time symmetry as regards past and future time. The expression for 

 ds^ is, under these conditions, of the form 



ds"" = - edr'' - e'{rdd'' + r sin^ Sd^^) + e'df 



where X, fx, v are functions of r, and t contains the unit-dimensional 

 constant c and is therefore to be measured in length units, one kilometre 

 corresponding to J- 10~^ seconds. 



A transformation of the radial co-ordinate leads to 



ds^ = - e\h' - (rMd^ + r' sin^ ddcf>') + e''dt\ 



Hence ^^ = ~e , g^^ = — >•^ .r/33 = — r^ sin' 6, g,^ = e', and g^^^ = 

 when a and ^ are unequal. 



The substitution of these values in the equations G^^ = leads to 



a determination of A, v, and ds is finally given by 



ds' = - y-^ dr' - rdff' - r' sin 'edcj)' + ydi\ 



where y = 1 — 2rn/r and ni is a constant of integration arising in the 

 solution of the differential equations. 



The differential equations of the path of a particle moving in the 

 gravitational field are found by applying the ordinary methods of the 

 Calculus of Variations to the equation S/ds = 0. 



