Page:StokesAberration1846.djvu/5

Theory of the Aberration of Light.

with the refracting surface travels along $AB$ with the

Fig. 2.

velocity $\mathrm {cosec} \ \psi \left\{V+q\sin(\psi +\alpha )\right\}$. Observing that ${\tfrac {q}{\mu ^{2}}}$ is the velocity of the æther within the refracting medium, and ${\tfrac {V}{\mu }}$ the velocity of propagation of light, we shall find in a similar manner that the lines of intersection of the refracting surface with the reflected and refracted waves travel along $AB$ with velocities

$\mathrm {cosec} \ \psi _{'}\left\{V+q\sin(\psi _{'}+\alpha )\right\},\ \mathrm {cosec} \ \psi '\left\{{\frac {V}{\mu }}+{\frac {q}{\mu ^{2}}}\sin(\psi '+\alpha )\right\}$

But since the incident, reflected and refracted waves intersect the refracting surface in the same line, we must have

$\left.{\begin{aligned}\sin \psi _{'}\left\{V+q\sin(\psi +\alpha )\right\}=&\sin \psi \left\{V+q\sin(\psi _{'}+\alpha )\right\},\\\mu \sin \psi '\left\{V+q\sin(\psi +\alpha )\right\}=&\sin \psi \left\{V+{\frac {q}{\mu }}\sin(\psi '+\alpha )\right\}.\end{aligned}}\right\}$

(A)

Draw $HS$ perpendicular to $AH$, $ST$ parallel to $NA$, take $ST:HS::q:V$, and join $HT$. Then $HT$ is the direction of the incident ray; and denoting the angles of incidence, reflexion and refraction by $\phi ,\phi _{'},\phi ^{'}$, we have

$\phi -\psi =SHT={\frac {ST\sin S}{SH}}={\frac {1}{V}}\times$ resolved part of $q$ along $AH$

$={\frac {q}{V}}\cos(\psi +\alpha )$. Similarly,

$\phi _{'}-\psi _{'}={\frac {q}{V}}\cos \left(\psi _{'}-\alpha \right),\ \phi '-\psi '={\frac {q}{\mu V}}\cos \left(\psi '+\alpha \right):$

whence

${\begin{aligned}\sin \psi =&\sin \phi -{\frac {q}{V}}\cos \phi \cos(\phi +\alpha ),\\\sin \psi _{'}=&\sin \phi _{'}-{\frac {q}{V}}\cos \phi _{'}\cos(\phi _{'}-\alpha ),\\\sin \psi '=&\sin \phi '-{\frac {q}{V}}\cos \phi '\cos(\phi '+\alpha ).\end{aligned}}$