• Here F is the equivalent focal length of the lens system.
• From equations (1) and (2), we get

• From figure

           O₂C = O₂P – CP                (∵  δ₁ = CP / BP)

∴         h₂ = h₁ – (BP) δ₁

Here  Δ = f₁ + f₂ − d, and it is known as optical separation.

• If P₁ and P₂ are the power of L₁ and L₂, and P is the power of lens system, then

           P = P₁ + P₂ – d P₁P₂

Position of second focal point (O₂F₂ = β₂)

• The real points from where the distances can be measured are O₁ and O₂
• The distance of F₂ will be measured from O₂,

           O₂F₂ = β₂

• From ΔM₂H₂F₂ and ΔCO₂F₂

But       h₂ = h₁ (1 – d / f₁)

Position of second principal point (O₂H₂ = α₂)

• The distance of second principal point is measured from the second optical centre O₂
• From figure,

           H₂O₂ = H₂F₂ – O₂F₂

or        α₂ = F – β₂             (∵  H₂F₂ = F and O₂F₂ = β₂)

• Since H₂ lies to the left of L₂

Position of first principal point (O₁H₁ = α₁)

• The distance of first principal point is measured from the first optical centre O₁

Position of first focal point (O₁F₁ = β₁)

• The distance of first focal point can be measured from the first optical centre O₁
• From figure

           O₁F₁ = H₁F₁ – O₁H₁             (∵    H₁F₁ = F and O₁H₁ = α₁)

or         β₁ = F – α₁

• Since F₁ lies to the left of L₁

To know in detail about cardinal points of a lens system in English click here and in Bilingual (Hindi/English) click here.