David Gerstman writes to Ask Mr. WeatherBlogger about rainbows:

"On Sunday we were headed back from the Catskills; south on 81 through Pennsylvania. Shortly after we passed Hazelton a storm passed us - traveling eastward. After we passed through the storm we could see the setting sun in the west and the overcast sky to the east.

"In similar circumstances at home those would be conditions to create a rainbow. But on Sunday we didn't see one. Why not? Other than sun shining on a passing rainstorm, what other conditions are necessary for a rainbow? - David."

Sunshine and rainfall are both necessary to produce a rainbow, David, but they aren’t, by themselves, sufficient. It’s all about angles, and French mathematician Rene Descartes figured it all out in 1637. Here's what I learned after poking into your question for a while:

Rainbows occur when sunlight strikes the spherical drops of rain as they fall from clouds. The light enters each drop and sort of ricochets around inside the droplet. It gets refracted, or bent, as it passes from air to water, reflected off the rear inside surface of the drop, then refracted again as it passes from water back into the air and back toward the observer. It emerges at an angle 42 degrees from the original path of the sunlight.

In addition, the refraction splits the white light into its various constituent frequencies, or colors, each of which bends at a slightly different angle – between 42 and 40 degrees. That’s what produces the concentric bands of color. We see just one color from each droplet, depending on our angle of view.

The size of the droplets, too, affects the colors we see. The smallest drops produce very white bows, including "fog bows." (Airline passengers can sometimes see such fully circular fog bows or "cloud bows" when they look out the window toward a cloud that’s in the opposite direction from the sun.) The largest droplets yield very red and yellow rainbows, weak in greens and blues.

Sometimes, some of the sunlight knocks around inside the droplet a second time, emerging at a slightly different angle, and we see a double rainbow.

What we actually perceive is the combined effect of light refracted and reflected from every droplet in the shower, and of the angles at which it returns to our eyes.

But to see any rainbow, the observer has to be exactly between the sun and the rain – facing what scientists call the "anti-solar point," or the spot in the sky directly opposite the sun.

The ‘bow" is actually the top portion of a circle with an angular radius from the observer of 42 degrees around that point. The Earth blocks the lower portion of the circle. At sunset, with the sun at the horizon, the top of the arc would be 42 degrees above the horizon.

I’m not sure why you didn’t see a rainbow on your trip south on I-81. There are a couple of possible answers.

If you’re not standing (or driving) directly between the sun and the rain shower, you will see nothing, or perhaps only a partial bow. Or, I would guess that if high clouds float between the sun and the shower, observers on the ground might be able to see both, but the direct sunlight may not reach the falling droplets. So, no rainbow.

That’s why rainbows (and consequently pots of gold) are so rare. And delightful.

You can read more about rainbows here. And here's a link to rainbow photos.