UPDATE The formula has changed as of the March 2011 patch, it is now
(city population * 1.1) + (capital population * .15) - 1 for each city connected to the capital, not including the capital.
- The Machu Pichu wonder increases the modifier by 20%, to
(city population * 1.3) + (capital population * .15) - 1.
- Arabia increases the constant by 1, to
(city population * 1.1) + (capital population * .15) (thanks WillfulWizard).
The economic overview also gives a far better explanation of how it is calculated - in other words, this question is now trivially solved by simply looking at that economic overview :)
Original answer below.
Okay, I did a bit more testing, and bwarner's answer is almost accurate:
Each city, excluding the capital, provides
(city's population * 1.25) + 0.01 gold per turn. Owning the Machu Pichu wonder increases the modifier by 20%, to
(population * 1.5) + 0.01, for all the cities.
The capital does not provide any gold.
The 1.25 and 0.01 can be seen in the
The economic overview also demonstrates these values.
I couldn't find any other factor which affects these values, and I checked different difficulty levels, different distances between cities, different city route type (road vs railroad, road/railroad vs harbor), different city health and whether it is occupied. Looks like it's solely the population.