Page:LorentzGravitation1916.djvu/38

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(63)

and if for the value (49) is substituted, this term becomes

or if in the latter summation is interchanged with and if the quantity

(64)

is introduced,

Finally, putting equal to zero the coefficient of each we find from (62) the differential equation required

(65)

This is of the same form as Einstein's field equations, but to see that the formulae really correspond to each other it remains to show that the quantities and defined by (63), f59) and (60) are connected by Einstein's formulae

(66)

We must have therefore

(67)

and for

(68)


§ 42. This can be tested in the following way. The function (comp. § 9, 1915) is a homogeneous quadratic function of the 's and when differentiated with respect to these variables it gives the quantities . It may therefore also be regarded as a homogeneous quadratic function of the . From (35), (29) and (32)[1], 1915 we find therefore

(69)

Now we can also differentiate with respect to the 's, while not the 's but the quantities are kept constant, and we have e.g.

(70)

According to (69) one part of the latter differential coefficient is

  1. The quantities in that equation are the same as those which are now denoted by .