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Hi Antonie

      I think I have understand your point and now I am replying to you with pretty much the same of what I have just replied to Lars, and hoping that you can find out what I have missed or misunderstood.  My result is 03:39:07, at a later westly-shifted pos: (47°10′N  9°0.3′E ) I too did not trust both altitudes and relied solely on the measured distance and a selected AP,as Frank himself has described the figure as being "accurate to the nearest tenth of a minute of arc." 

       I think using the traditional method based on different APs would yield slightly different rates and eventual UTs.

       There are the issues with assumed positions and LD rate of change,thanks to your detailed elaboration. You have come up with an LD rate of change of about 0.1′ per 32s at a predetermined AP, that is the averaged rate for a ten-minute interval between 03:35 and 03:45, obviously you re trying to get to the actual LD rate at the time of interest as close as possible. The LD rate of change itself is obviously irratic and would definitely vary between different positions and at different time(vary drastically if you re stationary or travelling due-westward at about 15° per hr. / as you have demostrated)Since Frank did not provide a set of specific coordinates, it's practically impossible to yield the specific LD rate of change. Everyone would come up with their own rate of change,which would only apply at their shifted assumed positions.

       If one were to use a rate of change that is directly based on lunar distance tables, then I think he has to understand that this rate only applies if an observer being physically at the geocentric point. (l used a rather coarse and straightforward hourly rate,between 0300 and 0400hrs,derived from Frank's LD table). 

       One way that will put yourself come close to the geocentric point condition is what you have described as being a "moving observer",one has to travel along your latitude line at an angular speed of about 15° per hour, by doing so, he would literately remain stationary with the geocentric point, all distant star positions and heights are now frozen. But the instanteneous and averaged LD rate would still vary from table to your location at the hemisphere,since the moon is extremely close to you than stars. The instantaneous rate would be greater if moon stays at your zenith,and slower if at your horizon.

      Therefore,If I were to use the LD rate from the LD tables or other means that were calculated from the geocentric point,the assumed position consequently has to shift westwardly with time. I have deduced a UTC of 03:39:07 base on :  (an LD hourly rate of 21.6′/ assumed time 03:37:00/assumed position 47°10′N  9°32′E). The position has to shifted westwardly to (47°10′N  9°0.3′E, travelling for 127 seconds at 0.25′/Sec) at 03:39:07 in order to best-suite that table-calculated rate.   Did not check the maps so I have no idea if the new location are still within Liechtenstein territory.