Page:PlummerAberration.djvu/3

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.