Page:StokesAberration1848.djvu/2

has not been made manifest in such fluids by any phænomenon hitherto observed? I have already attempted to offer an explanation on the latter supposition (Phil. Mag., volxxix. p. 6). Professor Challis, in his last communication, has considered the æther as an ordinary fluid.

In my paper last referred to, I have expressed my belief that the motion for which $udx+$ &c. is an exact differential, which would take place if the æther were like an ordinary fluid, would be unstable; I now propose to prove the same mathematically, though by an indirect method.

Even if we supposed light to arise from vibrations of the æther accompanied by condensations and rarefactions, analogous to the vibrations of the air in the case of sound, since such vibrations would be propagated with about 10,000 times the velocity of the earth, we might without sensible error neglect the condensation of the æther in the motion which we are considering. As far as the case in hand is concerned, Professor Challis might have regarded $\rho$ as constant, and treated $p$ as he has treated $s$. Suppose, then, a sphere to be moving uniformly in a homogeneous incompressible fluid, the motion being such that the square of the velocity may be neglected. There are many obvious phænomena which clearly point out the existence of a tangential force in fluids in motion, analogous in many respects to friction in the case of solids. When this force is taken into account, the equations of motion become (Cambridge Philosophical Transactions, vol. viii. p. 297)

${\frac {dp}{dx}}=-\rho {\frac {du}{dt}}+\mu \left({\frac {d^{2}u}{dx^{2}}}+{\frac {d^{2}u}{dy^{2}}}+{\frac {d^{2}u}{dz^{2}}}\right)$

(I.)

with similar equations for $y$ and $z$. In these equations the square of the velocity is omitted, according to the supposition made above, $\rho$ is considered constant, and the fluid is supposed not to be acted on by external forces. We have also the equation of continuity

${\frac {du}{dx}}+{\frac {dv}{dy}}+{\frac {dw}{dz}}=0$

(2.)

and the conditions, (1) that the fluid at the surface of the sphere shall be at rest relatively to the surface, (2) that the velocity shall vanish at an infinite distance.

For my present purpose it is not requisite that the equations such as (1.) should be known to be true experimentally; if they were even known to be false they would be sufficient, for they may be conceived to be true without mathematical absurdity. My argument is this. If the motion for which $udx+\dots$ is an exact differential, which would be obtained