December i, 1923] 



NATURE 



789 



Einstein considers the special case of slow motion in 

 a weak gravitation field, i.e. such that the metrical 

 tensor compouents g,.^ differ but little from their 

 Galileian values. Then, neglecting squares, etc., of 

 these small differences and also their derivatives with 

 respect to x,^ (quasi-stationary field), Einstein easily 

 obtains the Newtonian equations as a first approxima- 

 tion, with S2== -\c^g^^ as the classical potential of the 

 gravitation field. This treatment of the question is 

 repeated, so far as I know, by all exponents of 

 Einstein's theory. 



Now, as has recently occurred to me. the true 

 relation of Einstein's equations to those of Newton is 

 of a much more intimate nature, and remains valid, 

 no matter how strong the field and how much space 

 deviates from Euclidean behaviour. 



In fact, the frame most natural to adopt for an 

 interpretation of the complicated equations of motion 

 (i) of a particle being clearly its own rest-system, let 

 Xi, X2, X3 be the space-coordinates of the particle in 

 such a system (the latter, of course, to play its part 

 during an infinitesimal time and to be replaced success- 

 ively by others and others). Moreover, let for con- 

 venience the origin of x^, etc., be taken at the particle 

 itself. Then, at any instant, Xi = dXil(is = o{i->. =. 3), and 

 equations (i) will reduce to ds^=gitdxi' and the tliree 

 equations 



d / I dXi\ _ c* J 44\ 



'^^^ ~ n/^A »■ f' 



(2) 



where dt = dxjc, the fourth equation being already 

 utilised. Now, with i, k reserved for i, 2, 3, 



/44 \ =aikfiljjc _ 1 ^4\ .,„?£«« 



\i i ^ \cx, * aArJ^*^ dx,- 



The coordinates can always be chosen so as to make 

 ^i_^42_^i3_Q jj^jg means a frame not spinning 

 relatively to the stars. In these coordinates then, 

 or in such a rest-platform of the particle, 



„ d since the Xi can now always be measured along 

 'the principal axes of the operator or matrix ^** (when 

 also g" =ilgu), we have 



r44\= _ ^ ^« 



I i i 2gu ' CXi' 



no more to be summed over i, of course. These 

 values substituted in (2) give, with gu = - a^, and 

 since Xi =dXildt =0, 



d'( Jai^l_ c* eg. 



4.1, , 



(3) 



2 Jou'cx- 



Now, the space-line element of our platform being 

 dP =aiidxi* ■\-a^^x^ +«»»rf^»*, 



v'anrf^i.etc.are the length elements rft,, etc., measured 

 along the axes precisely as in (N), and the right-hand 

 member of (3) expresses the gradient of li= - ic*^«4 -f 

 const. With a proper choice of the constant, 



We thus see that, in the rest-system of the free particle, 

 the genera! relativistic equations (i) become identical 

 with the Newtonian equations of motion, rigorously, i.e. 

 whether the gravitation field is weak or not {zUjc* a 

 small fraction of unity or not), and no matter how 

 strongly the plat form -space differs from a homaloidal 

 (jr Euclidean space. 



This simple investigation is here given not merely 

 because it seems to put the general ecjuations (i) into 

 an interesting and familiar light, but also because it 

 vindicates the rights of the Newtonian equations of 

 motion. Lur>wiK Sii.hkrstkin. 



129 Seneca Parkway, Rochester, N.'N'., 

 September 19. 



NO. 2822. VOL. I 12] 



The Influence of Barometric Pressure on the 

 Specific Gravity of the Surface Water in Indian 

 Seas. 



It has for many years been recognised that any 

 alteration in barometric pressure over a wide expanse 

 of water produces concomitant changes in the surface 

 level, and Prof. J. W. Gregory [Scottish Geographical 

 Magazine, 1909, vol. xxv. p. 316), when discussing 

 the level of the sea, pointed out that " the sea in an 

 area beneath high air pressure has its surface pushed 

 downwards and the displaced water rises in the ad- 

 jacent areas." Since the waves of increased barometric 

 pressure occur at approximately the same time of 

 day in each degree of longitude, it follows that each 

 succeeding elevation and depression of the surface level 

 of the sea travels across the ocean like a wave from 

 east to west. In the region of India the barometric 

 pressure normally exhibits in every twenty-four hours 

 a double rise and fall with maxima at approximately 

 9.45 A.M. and 10.30 P.M. and minima at 3.30 a.m. and 

 4.30 P.M. 



Investigations of the specific gravity {(t^) of the 



Fi«. I. — Average specific gravity of the surface water and simultaneous 

 barometric pressures during a voyage from Bombay to the Andaman 

 Islands in October 1931. 

 The continuous line shows the specific gravity, and the dotted line the 

 barometric pressure, in each of the three figures. 



surface water of Indian seas have revealed a daily 

 double oscillation that occurs simultaneously with, 

 and must, I think, be due to, the alterations of 

 barometric pressure. This oscillation of specific 

 gravity is, however, only clearly seen in the open sea, 

 because in inshore waters it is obscured by other 

 changes due to tidal flow, etc. During the voyage 

 from Bombay to Port Blair, Andaman Islands, in 

 October 192 1, a four-hourly record of the specific 

 gravity of the surface water and the barometric 

 pressure was carefully kept, and the results obtained 

 are shown in Fig. i. This shows very clearly the 

 way in which, as the barometric pressure falls, the 

 specific gravity of the surface water rises, and vice 

 versa, the two curves alternating with one another. 



A variation in the specific gravity of the surface 

 water such as this might be due to (a) lateral hori- 

 zontal movements of m<i.sses of water, or (fc) an upwell- 

 ing of water from a deeper level. If the latter cause is 

 the true one, then the effect of changes in barometric 

 pressure shojild be found to depend on the relative 

 specific gravity of the surface water and of water 

 immediately underlying the surface layer. In October, 

 following on the effects of the south-west monsoon, 

 the upper-level water will be diluted and have a 

 lower specific gravity than that immediately below, 



