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For stars on the meridian, we can generally split things into cases above the pole and cases "below the pole" which is what you have here with your Kochab sight. The pole in question is the nearer celestial pole. In the northern hemisphere, that nearer pole, the north celestial pole, is a point on the sky about two-thirds of one degree from the North Star. In the southern hemisphere, the corresponding south celestial pole is a point in a nearly starless part of the sky. There are no bright stars at all in that star desert. The faint star σ Octantis is the closest southern analog to Polaris, and it is now known officially, by IAU declaration with a bit of comedy, as Polaris Australis. If you're in the northern hemisphere and you observe a meridian altitude lower than Polaris, like your Kochab example, that's "below the pole".  

Above the pole:
This category includes nearly all meridian sights where the celestial object reaches a maximum altitude during its daily motion. That includes LAN sights —the Sun at local noon— as well as less common cases like bright stars on the meridian (surprisingly rare) and also Venus in daylight on the meridian, which will be possible again for about six months starting this summer. It can also work for the Moon on the meridian but there are some tricky details, so that's best set aside.

All "above the pole" cases have that property that we're dealing with a maximum daily altitude, and they also have the property of "hour angle" equal to zero.  Below the pole meridian altitudes by contrast are cases where a star or other celestial body (even the Sun in very high latitudes!) reaches a minimum altitude for the day, "bottoming out". All the above the pole cases can be handled by one easy rule:
   Lat = Z.D. + Dec,
where Lat is latitude and can be positive or negative (negative is "South" latitude). The Z.D. is the zenith distance equal to 90°-Ho where Ho is the corrrected altitude. And Dec is the celestial body's declination, which, like latitude, can be positive or negative (negative for "South"). This covers all cases so long as we add one other "south is negative" rule: if your shadow points South, manually make the Z.D. negative. This is part of the observation process: when you take a Noon Sun sight in the tropics especially, pay attention to which way you're facing. Naturally, you won't get a shadow from a bright star, so in that case, use your imagination (if you're facing north when you shoot a star on the meridian, the "shadow" cast by it, if visible, would be pointing south).

Below the pole:
It's a different rule entirely here, but it "resembles" the usual Noon Sun rule. For a star on the meridian below the pole:
   Lat = P.D. + Alt,
where Lat in this case will always come out positive (see penguins? Make Lat negative). The P.D. is the polar distance equal to 90°-Dec (really 90°-|Dec| if we want the math to include the southern hemisphere case). And Alt here is the usual corrected altitude, or Ho.

Your Kochab sight:
Altitude 31°38' with 16 feet height of eye. The altitude correction in minutes of arc (to be subtracted) is very nearly
  √(ht in feet)+1/tan(Alt) = 5.6'.
That's close enough to 6'. We subtract that and get 31°32. That's the 'Alt' we need in Lat = P.D. + Alt. 

For the declination of Kochab, we need an almanac source. I find 74°02', near enough. Subtract that from 90° to get the P.D. That's 15°58' or better yet 16° -02'. Now I add that onto the Alt we just calculated: 31° 32' + 16° -02'. And finally we have the latitude 47°30'. Does that sound about right?

So how do we remember this "below the pole" rule?! Easy. Just draw a little "cartoon" of the sky facing north (for northern hemisphere) observers showing the North Star and another star below it. It could be Kochab... Let's go ahead and label it Kochab, but we'll know that it could be some other star below the pole. In your cartoon, include a horizontal line for the horizon. Draw in a vertical arrow up from the horizon to Kochab. That's "Alt". Also draw an arrow up from the horizon to the North Star (or slightly offset from it if you want to be careful and indicate that the north celestial pole is slightly offset from Polaris. That's your "Lat" (assuming you can remember that the latitude of Polaris is nearly equal to your latitude, then that's a known thing). And finally draw an arrow from Kochab up to the pole. That's the polar distance of Kochab, and here you have to remember that polar distance is 90-Dec. Clearly in this simple drawing, the length of the latitude arrow equals the length of the altitude arrow plus the length of the P.D. arrow. And that immediately implies the formular: Lat = P.D. + Alt. Important note on this cartoon drawing: I am not suggesting that you try to re-derive the equation like a 21st century Euclid every time you use it. That's a path to insanity and a guarantee that you will avoid meridian sights. The cartoon diagram is a mnemonic, a quick memory aid. 

As for longitude... For an "above the pole" meridian sight, like typical Noon Sun, you and the celestial body are on the same longitude if the timing is absolutely perfect (it's never perfect, but for the moment, let's pretend...). Since you are on the same longitude, you can look up the longitude of the star or Sun or other celestial body at that moment of time. At the time you gave, the GHA of Kochab was 303°21'. With that longitude, you're in Kazakhstan between the northern reaches of the Caspian and Aral Seas... :) Oh, but one more trick! For a "below the pole" meridian altitude, you and the star are on opposite longitudes, separated by 180°. So we need to add or subtract 180° from the GHA of Kochab. That gives 123°21' (goodbye Kazakhstan). This appears to be about one degree of longitude west of the only portion of Puget Sound at that latitude with a water horizon to the north. That's about as good as you can expect for a longitude by this method. It's in the neighborhood in a global sense, but not really useful for navigation. This falls into a category I call "NCW Longitude". It's "Not Completely Worthless" but that's all.

Frank Reed

PS: The pair of general equations for meridian sights:
  Lat = ZD + Dec,
  Lat = PD + Alt,
above and below the pole respectively. Notice that if you expand ZD and PD, the comparison may feel more analogous:
  Lat = 90° - Alt + Dec,
  Lat = 90° - Dec + Alt.
If that helps, use it. If not, stick with the first pair. They're fundamental.