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

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 $$t + dt$$

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

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 $$t + d t$$ for $$t$$, and we have therefore for this equation,

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

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

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