The form of the equations (12.20) and (12.21) can be changed by multiplying equa-
tion (12.20) by cos βand multiplying equation (12.21) by cos α. The resulting equa-
tions are further simplified by using the results from equations (12.18) and (12.19)
to produce the equations
nacos xcos β=ngcos αcos βdα
dx (12 .22)
−nacos ξcos α=−ngcos αcos βdα
dx (12 .23)which must hold when y=ymin.
Addition of the equations (12.22) and (12.23) after simplification produces the
condition
cos xcos β= cos ξcos α (12 .24)Now square both sides of equation (12.24) and verify that
cos^2 xcos^2 β= cos^2 ξcos^2 α
(1 −sin^2 x)(1 −sin^2 β) =(1 −sin^2 ξ)(1 −sin^2 α)(1 −n^2 g
n^2 asin^2 α)(1 −sin^2 β) =(1 −n(^2) a
n^2 g
sin^2 β)(1 −sin^2 α)
(12 .25)
Expand the last equation from equation (12.25) and simplify the result to show
equation (12.25) reduces to the condition that
|sin β|=|sin α| (12 .26)under the condition that y =ymin. The equation (12.26) implies that under the
condition that y=ymin there must be symmetry in the light ray moving through the
prism. That is, the conditions
β=α=A
2and x=ymin +A
2(12 .27)