Exercise 4: Where is the double (secondary) rainbow?

The minimum of the deflection function for λ=400 nm (λ=650 nm) is δ=233.3∘ (δ=230.2∘).
These rays are directed back up into the sky and the ground-based observer won't see them.
However, if you realize that you have just computed the deflection angles for rays entering the bottom half of the raindrop, you realize that you must look (with the sun at your back and the distant raindrops in front of you) at an angle of 233.3∘−180∘=53.3∘,,(230.2∘−180∘=50.2∘) relative to the horizontal to see the bright violet (red) band of the rainbow.
So the red band of the secondary rainbow appears below the violet band. (It really does!).
Thus the rays producing the secondary rainbow enter the bottom half of the raindrops and the order of the color bands is reversed relative to the primary rainbow.
Comparing these angles to those from Exercise 3, we see that the secondary rainbow appears above the primary rainbow.
The reduced brightness of the secondary rainbow is due to the additional internal reflection and the corresponding loss of irradiance during that reflection.
It is posible to extend Exercises 3 and 4 to account for the change in intensity occurring for every refraction or internal reflection.
For fun, it must be noted that fascination with the double rainbow became an Internet meme in 2010 when a person living on the border of Yosemite National Park recorded his reaction to a spectacular double rainbow.
At the time of writing, the video can be seen at: https://www.youtube.com/watch?v=OQSNhk5ICTI).