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Stephen - I have taken many 3- and 4- star fixes using a bubble sextant and Pub249v1 using a single AP.

I noticed that Gary LaPook posted some links - if these include his links on using the Polhemus calculator, those will be very helpful for what you want.

The idea of the single AP is that you shoot 3 or 4 timed sights where one is at your Fix Time and the other two are increments of 4 minutes before or after.  For example, on a 4-star shot, I will shoot one star 8 minutes before the fix time, the next at 4 minutes before the fix time, the next at the fix time, and the last 4 minutes after the fix time.  The 4-minute interval provides me a chance to record the previous shot and then set up the sextant for the next shot with a minute or so to spare.  The interval doesn't have to be 4 minutes but if you are using a bubble sextant with a 2-minute averager, 4 minutes is a pretty good interval.

Each star has a rate of rise or fall called Motion of the Body.- you can find this in Table 2 in your Pub249v1 or on the calculator face of the Polhemus computer and is given in angle-minutes per time-minute.  Rising objects have a negative sign and falling objects have a positive sign.

You multiply this rate of rise/fall by the number of minutes ahead or behind the fix time.  So, for the first shot which occurs 8 minutes before the fix time, you multiply by +8; for the shot which occurs 4 minutes before the fix time, you multiply by +4; for the shot which occurs at the fix time, you multiply by 0; and for the shot that occurs 4 minutes after the fix time, you multiply by -4 (note the sign).

For aircraft, you would also want to know your speed/track so you can go to Table 1 in your Pub249v1 (and, also on the face of the Polhemus computer) to figure out the rate of rise/fall of the object caused by your motion over the face of the globe (Motion of the Observer).  If you are stationary, this is 0, of course.  And if you are not moving, you don't need to worry about Coriolus.  If you are on a boat, your speed is likely to be slow enough that MOO doesn't matter.

You take the result of your multiplications and apply (add, with the correct sign) that to the star listings in Pub249v1.  What this does is, in effect, emulate a Running Fix by shifting the LOP in time by 4 or 8 minutes.

You can plot your LOPs using standard To/From procedures from the same AP.

Note that you will also want to look at Table 5 in your Pub249v1 to correct for Procession and Nutation - remember that the Pub249v1 is epoch based so there is some variation from year to year.  You can either move your AP by the amount and direction in Table 5 or you can move the center of your position triangle by the amount and direction in Table 5.

As others have mentioned, this is using the concept of the Timed Sight - everything is precomputed and all you have to do is shoot the target stars at the right time.  Of course, you can take the sights first and then you can still do all this afterwards tho it takes away one of the great strengths of this method which is generating a fix quickly after the last shot.

I hope this helps.