Page:StokesAberration1848.djvu/3

from the common equations, were stable, the motion which would be obtained from equations (1.) would approach indefinitely, as $\mu$ vanished, to one for which $udx+\dots$ was an exact differential, and therefore, for anything proved to the contrary, the latter motion might be stable; but if, on the contrary, the motion obtained from (1.) should turn out totally different from one for which $udx+\dots$ is an exact differential, the latter kind of motion must necessarily be unstable.

Conceive a velocity equal and opposite to that of the sphere impressed both on the sphere and on the fluid. It is easy to prove that $udx+\dots$ will or will not be an exact differential after the velocity is impressed, according as it was or was not such before. The sphere is thus reduced to rest, and the problem becomes one of steady motion. The solution which I am about to give is extracted from some researches in which I am engaged, but which are not at present published. It would occupy far too much room in this Magazine to enter into the mode of obtaining the solution: but this is not necessary; for it will probably be allowed that there is but one solution of the equations in the case proposed, as indeed readily follows from physical considerations, so that it will be sufficient to give the result, which may be verified by differentiation.

Let the centre of the sphere be taken for origin; let the direction of the real motion of the sphere make with the axes angles whose cosines are $l,m,n$, and let $\nu$ be the real velocity of the sphere; so that when the problem is reduced to one of steady motion, the fluid at a distance from the sphere is moving in the opposite direction with a velocity $\nu$. Let $a$ be the sphere's radius: then we have to satisfy the general equations (1.) and (2.) with the particular conditions

$u=0,\ v=0,\ w=0,\ {\text{when}}\ r=a;$

(3.)

$u=-l\nu ,\ v=-m\nu ,\ w=-n\nu ,\ {\text{when}}\ r=\infty$

(4.)

$r$ being the distance of the point considered from the centre of the sphere. It will be found that all the equations are satisfied by the following values,

$p=\Pi +{\frac {3}{2}}\mu \nu {\frac {a}{r^{3}}}(lx+my+nz),$ $u={\frac {3}{4}}\nu \left({\frac {a}{r^{3}}}-{\frac {a^{3}}{r^{5}}}\right)x(lx+my+nz)+l\nu \left({\frac {1}{4}}{\frac {a^{3}}{r^{3}}}+{\frac {3}{4}}{\frac {a}{r}}-1\right)$

with symmetrical expressions for $v$ and $w$. $\Pi$ is here an arbitrary constant, which evidently expresses the value of $p$ at an infinite distance. Now the motion defined by the above expressions