NavList:

I solved the problem this way:

 

We know, for six observations, the UTC and the observed altitude, and the set and drift.

 

We have almanac predictions of of GHA and Dec for 1600 and 1700.  We assume linear variation in between.

 

(1) - From an assumed 1600 Lat and Long, we can calculate the delta Lat and delta Long for each observation using set and drift.

 

(2) - We now have, for each observation, Lat, Long, and LHA (GHA+Long), so we can calculate an Alt (and Az).

 

(3) - Now, for a starting 1600 Lat and Long, we calculate an altitude corresponding to each of the six observations.

 

(4) - We calculate an error as the square root of the sum of the differences between the observed and calculated squared.

 

(5) - We search for a Lat and Long which minimizes this error.  This search iterates until convergence, so starting Lat and

 

Long values don't effect the result.  For example, 35N and 65W still produce the given answer.

 

(6) - We now can calculate Alt, Lat, and Long at each of six observation times (as well as 1600 and 1700).  So, in fact we have our desired position result at this point.

 

Continuing to culmination and transit values:

 

(7) - We do a second order fit to based on UTC to the six calculated altitudes calculated based on these six Lat and Long values..

 

(8) - We set the derivative equal zero to find the UTC of culmination (max altitude).

 

(9) - We do a linear fit based on UTC to the six calculated LHA's (or Az) and solve for the LHA=0 (or Az=180) to find the time of transit.

 

EXCEL is your friend.  See attached.

 

Results:

 

                    UTC                  Alt                   Lat                Long

 

1600 fix        17:00:00        +71°  21.14'    +37° 13.76'  -70° 13.63'   (5)&(6)

 

transit          16:37:31.03    +73° 17.17  '  +37° 11.27'    -70° 12.34'    (9)

culmination  16:37:56.44    +73° 17.18'    +37° 11.24'    -70° 12.32'    (8)

dif tran-cul    -25.406 sec   -0.01505 min  0.0281 min    -0.01461 min

 

1700 fix      17:00:00.00 +72°  36.57'     +37°  09.78'    -70°  11.56'  (6)

 

All in fairly close agreement with the results of others.