i. Consider a principal section ABC of a prism of absolute refractive index n kept in air as shown in figure.
ii. Let A be the refracting angle of prism and surface BC be the base.
iii. A monochromatic ray PQ obliquely strikes first reflecting surface AB such that, angle of incidence ∠PQM at Q is i.
iv. After refraction at Q, the ray deviates towards the normal and strikes second refracting surface AC at R which is the point of emergence.
v. Angles of refraction at Q (∠NQR) and at R (∠QRN) are r1 and r2 respectively.
vi. After R. the ray deviates away from normal and finally emerges along RS making e as the angle of emergence.
vii. Emergent ray RS meets an extended incident ray QT at X if traced backward. In this case, ∠TXS gives the angle of deviation.
viii. From figure,
∠AQN = ∠ARN = 90°
∴ From quadrilateral AQNR,
A + ∠QNR = 180° ………. (1)
From ∆ QNR,
r1 + r2 + ∠QNR = 180° ………. (2)
∴ A = r1 + r2 ……… (3)
ix. Angle δ forms an exterior angle for ∆ XQR.
∴ ∠XQR + ∠XRQ = δ
∴ (i – r1) + (e – r2) = δ
∴ (i + e) – (r1 + r2) = δ
From equation (3),
δ = i + e – A
∴ i + .e = A + δ