Derive an expression for the focal length of the combination of two thin lenses when they are SEPERATED  by a small distance?

Consider two lenses L1 and L2 separated by a small distance 'd' apart, as shown below. A ray of light AB initially parallel to the principal axis hits the lens L1 and deviates and then hits lens L2.

here

f1 is the focal length of L1 and

f2 is the focal length of L2

and

δ1 is the deviation produced by L1

δ2 is the deviation produced by L2

so,

form simple geometry

δ1 = h1 / F1

and

δ2 = h2 / F2

now, 

total deviation of the ligth ray will be

δ = δ1  +δ2

or

δ =  (h1 / F1) +  (h2 / F2)

also

at lens 2 and triangle BCD

h2 = h1 - CD = h1 - BD.tanδ1

or

h2 = h1 - d.tanδ1

now as δ1 is very small, tanδ1 ~ δ1

thus,

h2 = h1 - d.δ1

so, from earlier relation

h2 = h1 - d.(h1/f1)

thus,

δ = δ1  +δ2 = (h1/f1) +  [(h1 - d.(h1/f1)) / f2]

or

δ =  (h1/f1) + (h1/f2) - (dh1 / f1.f2)

now,

for the combination of the two lenses let F be the combined focal length.

So, the total deviation will be given as 

δ =  h1 / F

or

 (h1/f1) + (h1/f2) - (d.h1 / f1.f2) = h1 / F

thus,

1/F = 1/f1 + 1/f2 - (d / f1.f2)

so, the combined focal length will be

F = f1.f2 / (f1 + f2 - d)