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quadratic form from which the gravitational equations are derived is the same as the quadratic form which determines the optical properties of the medium. Indeed, the example which I considered on p. 262 of my first paper would seem to indicate that this was not the case. It should be mentioned that in the first seven equations in this example there is a misprint, $\epsilon \mu$ should be replaced by $(\epsilon \mu )^{-1}$. On the above view Einstein's idea of an influence of gravitation on light is simply an hypothesis, but a very interesting and reasonable one. It may be remarked, however, that in the theory of surfaces there are two fundamental quadratic forms, and we may perhaps expect something similar in general relativity.

With regard to possible extensions of the idea of relativity it may be worth while to consider transformations analogous to the contact transformations of dynamics in which the co-ordinates x, y, z, t and the component velocities u, v, w correspond to a new set $\left(x_{1}y_{1}z_{1}t_{1}u_{1}v_{1}w_{1}\right)$ in such a way that the differential equations

${\frac {dx_{1}}{u_{1}}}={\frac {dy_{1}}{v_{1}}}={\frac {dz_{1}}{w_{1}}}=dt_{1}$

are a consequence of the equations

${\frac {dx}{u}}={\frac {dy}{v}}={\frac {dz}{w}}=dt$

This may be secured by making a single quadratic form, such as

${\begin{array}{cc}\left(dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}\right)\left(c^{2}-u^{2}-v^{2}-w^{2}\right)\\+\left(c^{2}dt-udx-vdy-wdz\right)^{2},&\left(c^{2}>u^{2}+v^{2}+w^{2}\right)\end{array}}$

an invariant^[1]. Various other quadratic forms consisting of sums of squares may, of course, be adopted instead.

H. Bateman.

Throop College.

Pasandena, Cal.

Aug. 10th, 1918.

1. This is a positive definite quadratic form in the variables dx - udt, dy - vdt, dz - wdt, and so can only vanish when all these quantities are zero.