Page:PlummerAberration.djvu/3

254 where A1F1 = f, A1C = b. If we admit a contraction λ-1 which will make CF1 = λ.CF2, the equation of BA2B' becomes

$z^{2}+y^{2}=4f\lambda(x+b/\lambda)$.

This is a real deformation of the figure of the mirror. But if BA3B' is the surface at which the advancing wave-front actually meets the moving mirror

$A_{2}A_{3}/v=A_{3}C/U=A_{2}C/(v+U)$

where v is the velocity of the mirror, and the virtual surface on which the wave falls becomes

$z^{2}+y^{2}=4f\lambda(1+v/U)\{x+b/\lambda(1+v/U)\}$.

This is still a paraboloid, and the wave BCB' will reach its focus F3 after a time

$t=f\lambda(U+v)/U^{2}+b/\lambda(U+v)$.

Hence no change of focus will be detected provided F2F3 = vt. Now

$\begin{array}{l} CF_{3}=f\lambda(1+v/U)-b/\lambda(1+v/U)\\ \\CF_{2}=f/\lambda-b/\lambda\end{array}$

so that

$F_{2}F_{3}=f\lambda(1+v/U)-f/\lambda+bv/\lambda(U+v)$

which is equal to vt if

$f\lambda.v(U+v)/U^{2}=f\lambda.(U+v)/U+f/\lambda$

or

$\lambda^{-2}=(U+v)\left(1/U-v/U^{2}\right)=1-v^{2}/U^{2}.$

This is the law of contraction previously obtained, and the result is a complete compensation of the optical effect due to the motion of the mirror.