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of Fizeau's water-tube experiment, the prediction of the law connecting electronic mass with velocity, and the prediction of ponderomotive electromagnetic forces in moving media.

One final, and therefore crucial, test remains: gravitation. It was soon noticed that the hypothesis was inconsistent with the exact truth of Newton's gravitational law of force $mm'/r^{2}$. Thus the hypothesis of relativity predicts that a freely moving planet cannot describe a perfect ellipse about the sun as focus. This prediction is made on quite general grounds, just as the conservation of energy predicts that a stream of water cannot flow uphill. But the conservation of energy by itself is powerless to predict what will be the actual course of a stream of water, and in precisely the same way the hypothesis of relativity alone is powerless to predict what will be the orbit of a planet. Before this or any other positive gravitational predictions can be made, additional hypotheses must be introduced. The main trunk of the tree is the relativity hypothesis already mentioned; these additional hypotheses form the branches. The trunk can exist without its branches, but not the branches without the trunk. Whether the branches have been placed on the trunk with complete accuracy is admittedly still an open question—it must of necessity remain so until the difficult questions associated with the gravitational shift of spectral lines have been finally settled—but the main trunk of the tree can be disturbed by nothing short of a direct experimental determination of the absolute velocity of the earth, and the only means which can possibly remain available for such a determination now are gravitational.

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The Michelson-Morley Experiment and the Dimensions of Moving Bodies.

By Prof. H. A. Lorentz, For.Mem.R.S.

AS doubts have sometimes been expressed concerning the interpretation of Prof. Michelson's celebrated experiment, some remarks on the subject will perhaps not be out of place here. I shall try to show, by what seems to me an unimpeachable mode of reasoning, that, if we adopt Fresnel's theory of a stationary æther, supposing also that a material system can have a uniform translation with constant velocity $v$ without changing its dimensions, we must surely expect the result that was predicted by Maxwell.

Let us introduce a system of rectangular axes of co-ordinates fixed to the material system, the axis of $x$ being in the direction of the motion. Then, with respect to these axes, the æther will flow with the velocity $-v$. The progress of waves of light, relatively to them, may be traced by means of Huygens's principle; for this purpose it suffices to know the form and position of the elementary waves. For the sake of generality I shall suppose the propagation to take place in a material medium of refractive index $\mu$, so that, if $c$ is the velocity of light in the æther, the velocity in the medium when at rest would be $c/\mu$. The elementary wave formed in the time $dt$ around a point $P$ will be a sphere of radius $(c/\mu )dt$, of which the centre $P'$ does not, however, coincide with $P$, the line $PP'$ being in the direction opposite to that of $OX$. and having the length $\left(v/\mu ^{2}\right)dt$ (Fresnel's coefficient).

If $Q$ is any point on the surface of the sphere, $PQ$ can be considered as an element of a ray of light, and $w=PQ/dt$ will be the velocity of the ray. Confining ourselves to terms of the second order, i.e. of the order $v^{2}/c^{2}$, and denoting by $\delta$ the angle between the ray and $OX$, we have

${\frac {1}{w}}={\frac {\mu }{c}}+{\frac {v}{c^{2}}}\cos \delta +{\frac {v^{2}}{2\mu c^{3}}}\left(1+\cos ^{2}\delta \right)$

(1)

Now, let $A$ and $B$ be points having fixed positions in the material system. The course $s$ of a ray of light passing from $A$ to $B$ will be determined by the condition that the integral

$\int {\frac {ds}{w}}$

(2)

is a minimum. Using the above value of $1/w$, it is easily shown that, if quantities of the second order are neglected, the course of the ray is not affected by the translation $v$, so that, if $L_{0}$ is the path of the ray in the case $v=0$, and $L$ the real path, these lines will be distant from each other to an amount of the second order only. Hence, if in the case of a translation $v$ we calculate by means of (1) the integral (2), both for $L$ and $L_{0}$, the two values will differ by no more than a quantity of the fourth order; indeed, since the integral is a minimum for $L$, the difference must be of the second order with respect to the distances between $L$ and $L_{0}$, and these distances are already of the second order of magnitude.

It is seen in this way that, so long as we neglect terms of an order higher than the second, we may replace

$\int \limits _{L}{\frac {ds}{w}}$ by $\int \limits _{L_{0}}{\frac {ds}{w'}}$

an argument that must not be overlooked in the theory of the experiment. On the ground of it we shall commit no error if, in determining the paths $L_{1}$ and $L_{2}$ of two rays that start from a point $A$, and are made to interfere at a point $B$, we take no account of the motion of the apparatus. The change in the difference of phase produced by the translation will be given by the difference between the values which the integral

$\int {\frac {v^{2}}{2\mu c^{3}}}\left(1+\cos ^{2}\delta \right)ds$

takes for the lines $L_{1}$ and $L_{2}$ so determined. If, along the first of them, $\cos ^{2}\delta =1$, and along the

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