OCTOBEK 29, 1920] 



SCIENCE 



413 



the Gulf States and the two IsTavy stations 

 at Colon and Santo Domingo, will form a 

 " network which, it is believed, will furnish 

 information of great value in the study of 

 these destructive storms and in forecasting 

 their direction and rate of movement." 

 Whether or not hurricanes occur, observations 

 will be made twice daily and the data on 

 trades, antitrades, etc., will well repay the 

 effort, for very little is known of the winds 

 aloft in those regions. If funds permit, this 

 program will be extended during the next 

 several years. 



Not only is it essential that means be pro- 

 vided for the extension of pilot balloon work 

 in the West Indies, but also in the United 

 States proper. At present there are about 

 two dozen stations sending daily reports of 

 free-air wind conditions to the forecast centers 

 of the Weather Bureau. This information 

 forms the basis of forecasts that are issued 

 for the information of aviators in the Aerial 

 Mail Service and the Army and ISTavy Air 

 Services. At least fifty, and preferably a 

 hundred, additional stations are needed. It 

 would be possible, with such a net-work, to 

 construct upper-air wind charts from which 

 accurate and detailed forecasts could be made. 

 It is to be hoped that Congress will see the 

 importance of providing this additional equip- 

 ment, for its installation would find a direct 

 and immediate reflection in the increased 

 safety of aviation, and in the increased effi- 

 ciency of our aerial services. 



0. LeEoy Meisinger 



Washington, D. C. 



SPECIAL ARTICLES 

 NOTE ON EINSTEIN'S THEORY OF GRAVITA- 

 TION AND LIGHT 



This paper contains a statement of some 

 apparently unnoticed results dealing with 

 light rays and orbits in Einstein's general 

 theory of gravitation. The full proofs will 

 be published in the mathematical journals. 



We recall briefly that Einstein, in his gen- 

 eral relativity theory, introduces ten potential 

 functions gij^ (in contrast with the single 

 function api)earing in the liTewtonian theory) ; 



these are the coefficients in the fundamental 

 quadratic form 



rfs^ = Sgitdxidxi,, 



which defines the four-dimensional space-time 

 world (x^x„x^xj. When there is no actual 

 gravitation the manifold can be written in 

 the euclidean form dx^^ + dx^ -f dx^ + dx^, 

 or dx^ -\- dy^ -\- dz- — dt^ in the usual coordi- 

 nates. The path of a free particle is then 

 straight, and so is the path of a light pulse. 

 In the general gravitational case, the ten 

 potentials obey (in space not occupied by 

 matter) a certain set of ten difl^erential 

 equations of the second order i?j^.=;0, where 

 the left-hand members are the components of 

 what is known in the literature as " the con- 

 tracted Eiemann-Christoffel curvature tensor " 

 (Why not call it simply the Einstein tensor?). 

 A free particle then describes a geodesic, or 

 path of minimum lengtli s. Light rays are 

 found by adjoining the condition that ds 

 vanishes. When the quadratic form is put 

 equal to zero, the result will be described as 

 the light equation. 



I. Our first result is that if an Einstein 

 manifold has straight geodesies it is neces- 

 sarily euclidean. This means that if, in an 

 unknown field with vanishing Einstein tensor, 

 coordinates can be introduced such that the 

 paths of all particles are expressible by linear 

 equations, then the field is free from gravita- 

 tion. It is to be noted that curved four- 

 dimensional manifolds with linear geodesies 

 exist: but oiu- result shows that they do not 

 obey Einstein's equations. 



II. An analogous result holds for light rays. 

 If in an unknown Einstein field four coordi- 

 nates can be introduced so that the light 

 equation takes the usual form dx~ -\- dy^ -\- 

 dz^ — dt^ = 0, then there is no gravitation 

 (that is, the manifold is euclidean). This 

 requires proof since an arbitrary function may 

 be introduced as factor in the first member 

 without changing the light equation, although 

 this in general changes the field and the 

 geodesies. 



III. We pass now to general manifolds 

 where the paths can not be regarded as 



