Lensmaker's equation when the 2 mediums on both sides of the lens are different

[phantomvommand]

Hi, I chanced upon an interesting diagram lately, and was wondering what the lensmakers equation for such a lens would be. Attached below is the diagram. Basically, the refractive indexes of the 2 mediums surrounding each side of the lens are different.

They have also calculated the focal distances as shown below. However, I notice some inconsistency. f1 seems not to involve the refractive index of the opposite medium, while f2 does. Is the calculation of focal length correct, and what is the lensmakers equation when the mediums around the lens are different?

 

[Charles Link]

I verified the expression for $f_2$. To do this from first principles, (which I did), you have parallel rays incident from the left, and compute the refraction at the first surface for a height $h$ using Snell's law with paraxial approximations. Next compute the refraction at the second surface, and finally compute the focal length, where the parallel rays converge on axis, independent of $h$.
It would be a similar calculation for $f_1$.
I think the textbook is correct.
Note: To compute the angle $\phi$ of the normal of the curved surface as a function of $h$, you can use
$\frac{d \phi}{dh} \approx \frac{1}{R}$, so that $\phi=\frac{h}{R}$ in the paraxial ray aprroximation. (Traveling on a circle, $\frac{d \phi}{ds}=\frac{1}{R}$, where $s$ is the arc length. Near the optic axis, $h \approx s$).