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Adrian F,
Many thanks for your interesting answer. I was going to believe that "nobody seems interested in trying to decode the numbers". For the (cumulative) errors on 15 September Greenwich mean noon, you arrive at exactly the same results as I do. When it comes to the daily rates, there you have slightly lower values than I arrived at. I got 0.9^s losing  for #1969, and 2.3^s losing for #5774, per day, constant rate.

I see in your graphs C how you arrived at your values. You have identified a trend of slightly decreasing rate with time and extrapolated that trend to 15 September. A quadratic behaviour. I don't think it is possible to conclude with certainty such a trend from the given data. As the chronometer readings are on integer, or half, seconds only I guess most of the noise in the data is due to measurment uncertainties. Using a quadratic rate will make the daily reductions of observations more difficult as well. The easiest solution is just to use the first and last points. For #1969 you then get 24^s/26^d = 0.92^s; for #5774 you get 59^s/25^d=2.36^s, per day.

I myself made a linear least squares fit of the data, and took the slope of that as rate. With two decimals I got 0.91^s and 2.34^s, respectively.

It may look cumbersome to manually make a linear least squares fit, but if you center the data around zero and use integer values it is no large work. Attached my solutions. For #1969 I centered the dates, x, around 1 September, and the errors, y, around 2^m50^s. Aim at getting sum(x) or/and sum(y) zero or as close as possible. This will lighten the following calculations. Then you have to find y² and x·y and sum those columns as well. 

As an extra, I also calculated y² and then found the squared correlation coefficient, to see which chronometer was the best one. Or at least had the best fit. They were very nearly equal with #5774 slightly better. But this was just extra fun and not necessary for solving the problem.

Regarding the 1 p.m. drop time I mentioned, that was just a guess. I calculated the resulting longitude and looked at Google Earth and found it probable. I read somewhere that the time ball was situated on the Customs house on Prince William Street at the guessed era.

The language in the log is Swedish, but the month's names are equal or nearly equal to the English ones.
Lars