Finding The Latitude
If a star is at your Zenith, its declination is equal to your latitude. If the star is north or south of the Zenith, its declination is larger or smaller than your latitude, and the latitude is equal to the star's declination plus a correction for the star's zenith distance (90°  the star's altitude):
your latitude = the star's declination ± its zenith distance
= the star's declination ± (90°  the star's altitude)
If the star is north of our Zenith, its declination must be too large to be equal to our latitude, so we subtract the zenith distance to correct for that. If the star is south of our Zenith, its declination must be too small to be equal to our latitude, so we add the zenith distance instead.
EXAMPLE 1: Star South of the Zenith (Azimuth = 180 degrees)
What is our latitude if we observe Sirius crossing our Meridian at an altitude of 51 degrees, and an azimuth of 180 degrees?
The declination of Sirius is 17° (from the text), and its zenith distance is (90°  the altitude) = 90°  51° = 39°.
Since it is South of the Zenith, we add the zenith distance to the declination to find our latitude:
our latitude = the declination + the zenith distance = 17° + 39° = +22° (North latitude).

EXAMPLE 2: Star North of the Zenith (Azimuth = 0 degrees)
What is our latitude, if we observe Sirius crossing our Meridian at an altitude of 51 degrees, and an azimuth of 0 degrees?
The declination of Sirius is 17° (from the text), and its zenith distance is (90°  the altitude) = 90°  51° = 39°.
Since Sirius is North of the Zenith, we subtract the zenith distance from the declination:
our latitude = the declination  the zenith distance = 17°  39° = 56° (South latitude). 
Finding the Longitude:
In time units, the difference between two longitudes is the same as the difference in their Local Sidereal Times:
Longitude 1  Longitude 2 = Local Sidereal Time 1  Local Sidereal Time 2
If we use the Prime Meridian through Greenwich as the second longitude, and define Greenwich Sidereal Time (GST) as the Local Sidereal Time there, we get (reversing the order, so that positive results correspond to West longitudes)
(W/E) longitude  0 = (+/) (Greenwich Sidereal Time  Local Sidereal Time)
For Meridian observations of a star, the Local Sidereal Time is the same as the star's right ascension:
(W/E) longitude = (+/) (GST the star's Right Ascension)
In other words, to find our longitude we need only note the GST at which a star crosses our Meridian, and subtract the star's right ascension from that. If we happened to have a watch that keeps GST this would be simple (there used to be ads for such watches in astronomy magazines, as mechanically driven watches can be "regulated" to run fast or slow; but "quartz" watches cannot be adjusted in this way). But with a normal watch we have to calculate the GST.
To do this, we convert the local Standard/Daylight time to Greenwich Mean Time (GMT), also called Universal Time (UT), by adding or subtracting the whole number of hours from our standard meridian to the Prime Meridian; several tables in the text (e.g., Tables 14 and 24) show conversion values for various time zones. For Pacific Standard Time we add 8 hours; for Pacific Daylight Time (which is already an hour ahead) we add 7 hours.
Universal Time is a solar time; we need Greenwich Sidereal Time. To find this, we need to find the difference between solar and sidereal times at the time of the observation. Appendix 12 (in the text) is a tabulation of this difference at midnight on each day throughout the year (strictly speaking, it is a tabulation for 2009; but the notes at the end of the table allow corrections for other years). We move all the way around the Sun, causing it to move through 24 hours (1440 minutes) of Right Ascension in one year (365.25 days), or about 4 minutes per day. As a result, the tabulated difference usually increases by 4 minutes each day (because of accumulated roundoff errors, the amount sometimes changes by only 3 minutes).
Because the tabulated value is different from one day to the next, there is a gradual change in the quantity during the day. This change in the value is called the acceleration of time. To take it into account, we round off the UT to the nearest quarter of a day, then add a oneminute correction for each quarter of a day after midnight.
With all this in mind, the GST at the time of your observation is
GST =  UT at the time of transit (calculated from your clock time/zone)
+ difference between GST and UT at midnight (from Appendix 12)
+ interpolation correction for the acceleration of time 
After finding the GST you can compare it to (subtract it from) the star's right ascension, and obtain the longitude in time units. Since the Earth rotates to the East, the local clock time is always later to the East. So if the GST is bigger than the star's right ascension, it is later at Greenwich, and you must be West of Greenwich; while if the star's right ascension is bigger, it is later at your longitude, and you must be East of Greenwich.
Of course we do not normally express longitudes in time units, so the last step is to convert the result to degrees, using the Earth's rotation rate, 360 degrees in 24 sidereal hours, which is equal to 15 degrees in 1 sidereal hour, or 1 degree in 4 sidereal minutes. This requires dividing the minutes by 60, adding the result to the hours, then multiplying that result by 15.
EXAMPLE: What is our longitude, if we observe Sirius crossing our Meridian at 8:15 pm (PST) on December 22, 2010?
Convert Local Time (PST) to Universal Time (UT)
Start with the local time: = 8:15 pm PST (Dec 22)
Add 12 hours to convert to 24hour time: = 20:15 PST (Dec 22)
Add 8 hours to convert to UT: = 28:15 (Dec 22)
Since there are only 24 hours in a day, add a day and subtract 24 hours:
= 4:15 (Dec 23) 
Find the difference between solar and sidereal time (from Appendix 12)
Look up table value for Dec 23, 2009: = 6:06
Consult notes at end of table for correction to 2010: =  1 minute
Add correction to table value: = 6:05
 
Calculate the acceleration of time (the interpolation from midnight UT to the actual UT)
Take the UT: = 4:15
Round to the nearest 6 hours: = 6:00
For each 6 hours, use 1 minute for the acceleration of time: = 1 minute
 
Calculate the Longitude in Time Units
Add the three terms to get the GST: = 4:15 + 6:05 + 0:01 = 10:21
Subtract the Right Ascension of the star from the GST to get the Longitude in time units: = 10:21  6:45 = 3:36
 
Convert the Longitude to degrees
Divide minutes by 60 (min/hr) = 36/60 = .6
Add that to the hours = 3.6
Multiply that by 15 (° /hr): = (3.6 hrs) x (15° /hr) = 54° (west, since positive)


