Page:StokesAberration1845.djvu/3

Mr. G. G. Stokes on the Aberration of Light.

which corresponds to the point ($x,y,z$) in its front at the time $t$, we have

$x'=x+\left(u-V{\frac {d\zeta }{dx}}\right)dt,\ y'=y+\left(v-V{\frac {d\zeta }{dy}}\right)dt,\ z'=z+(w+V)dt;$

and eliminating $x,y$ and $z$ from these equations and (1.), and denoting $\zeta$ by $f(x,y,t)$, we have for the equation to the wave's front at the time $t+dt$

$z'-(w+V)dt=C+Vt+f\left\{x'-\left(u-V{\frac {d\zeta }{dx}}\right)dt,\ y'-\left(v-V{\frac {d\zeta }{dy}}\right)dt,t\right\}$

or, expanding, neglecting $dt^{2}$ and the square of the aberration, and suppressing the accents of $x,y$ and $z$,

$z=C+VT+\zeta +(w+V)dt$

(3.)

But from the definition of $\zeta$ it follows that the equation to the wave's front at the time $t+dt$ will be got from (1.) by putting $t+dt$ for $t$, and we have therefore for this equation,

$z=C+VT+\zeta +\left(V+{\frac {d\zeta }{dt}}\right)dt$

(4.)

Comparing the identical equations (3.) and (4.), we have

${\frac {d\zeta }{dt}}=w$

This equation gives $\zeta =\int wdt$: but in the small term $\zeta$ we may replace $\int wdt$ by ${\tfrac {1}{V}}\int wdz$: this comes to taking the approximate value of $z$ given by the equation $z=C+Vt$, instead of $t$, for the parameter of the system of surfaces formed by the wave's front in its successive positions. Hence equation (1.) becomes

$z=C+Vt+{\frac {1}{V}}\int wdz$

Combining the value of $\zeta$ just found with equations (2.), we get, to a first approximation,

$\alpha -{\frac {\pi }{2}}={\frac {1}{V}}\int {\frac {dw}{dx}}dz,\ \beta -{\frac {\pi }{2}}={\frac {1}{V}}\int {\frac {dw}{dy}}dz$

(5.)

equations which might very easily be proved directly in a more geometrical manner.

If random values are assigned to $u,v$ and $w$, the law of aberration resulting from these equations will be a complicated