ESSAYS 655 



The most genera possible transformation to coordinates x h x t , x 8 , x 4 will give for 

 ds an expression of the form 



ds* ~g\\dx\ +gsadx? 2 +gux\ +j?ux* + 2gvtdx\dx t + 2g u dx 1 dx t + 2g u dr l dx i + 

 igiartx \dx% + 2g<ndx<idxi + 2g u dxtdx^ 



in which the ^s are functions of the coordinates, and completely specify the trans- 

 formation. The values of the ^s were so determined as to reduce the gravitational 

 field at P to zero, and then, equating the two values of ds\ a relation was obtained 

 which specified the gravitational field in so far as to show how it might be reduced 

 to zero. Out of all the possible groups of ten equations to determine the gs, 

 Einstein, by means of the Riemann-Christoffel theory of tensors, succeeded in 

 picking out the few covariant groups, and amongst these he found one, and one 

 only, which could be reduced to a form satisfying approximately the equation 

 V 2 F=o, Laplace's equation for the Newtonian gravitation potential at a point in 

 ordinary space where there is no matter, when g ti was written for V. In the 

 presence of matter, the equation to be satisfied would be V*^= -4nG i p, where p 

 is the density of the matter and G is the gravitation constant ; and the generalised 

 form of this was found by Einstein in a certain expression T which occurred in the 

 group of ten equations for the ^'s, which could be taken as representing the energy 

 of the gravitational field, and that when this was done, the ten equations satisfied 

 Hamilton's principle. T is not a tensor, but the electromagnetic energy is, and 

 consequently also satisfies Hamilton's principle, from which it follows that in the 

 presence of an electromagnetic field of energy E, the total energy will be T+ E. 



Lorentz has approached the subject directly from the vector geometry of the field- 

 figure, the indicatrices in the case of Einstein's world being conjugate hyperboloids 

 with one real and three imaginary axes. He first expresses a system containing 

 material particles and one electromagnetic field in terms of a single function //to 

 which Hamilton's principle can be applied directly. H consists of three parts, 

 relating to the material particles, the electromagnetic field, and the gravitation 

 field respectively. Then, by the introduction of coordinates, Einstein's equations 

 are obtained as a direct consequence of the application of Hamilton's principle 

 to the function H : for the most general form of electromagnetic system : for a 

 gravitation-field consisting of incoherent similar material particles, with or without 

 molecular forces, acting between them, in such manner that the system would 

 be regarded as possessing a potential energy depending on the density only : in a 

 more general case, applicable, e.g. to systems which are non-isotropic as regards 

 both configuration and molecular action. To arrive at Einstein's complete 

 equations he has to add the term which in Einstein's equations refers to thermo- 

 dynamic and other effects, which in so far as they are not conservative, cannot 

 legitimately be treated by Hamilton's principle. 



The points of Lorentz's field-figure will be identical with the points of Einstein's 

 selected form of the time-space, in which the passage from point to point is effected 

 by pure rotations, if the axes be chosen so that dx h dx^dx 3 intersect the conjugate 

 indicatrix, of which the semidiameters, determining the units of length, have the 

 length ie, making g n , g-n, gst negative, while dx t intersects the indicatrix, the 

 semidiameter of which is e, so that gu is positive. It follows that any deformation 

 of the field-figure will leave unchanged the lengths of lines so expressed— in 

 natural measure— so that in the geometrical representation, the covariance of the 

 equations, when coordinates are introduced, is assured beforehand. 



To determine the ten g's for any point instant other than the selected orig 

 of coordinates for which the gravitation field vanishes, we have ten differential 



