If a locomotive is passed from one railroad to another, the clock starts when it moves across the boundary between the two railroads. The owning railroad then accumulates the time owed until the locomotive returns. The amount owed can be paid off by when the second railroad sends one of its own locomotives to the first for an equivalent length of time, or by a cash payment, if things get really out of balance.
If the locomotives exchanged are of different horsepower ratings, then the time owed is pro-rated for the horsepower difference.
If the second railroad passes the locomotive in turn to a third, the second and third railroads maintain an account of the horsepower-hours owed to the second, even if it isn't their own locomotive. As for the owning railroad, they simply keep accumulating hours owed from the second.
1) Rented or leased. This does happen every once and a while when one road is power short. Railways lease engines from various leasing outfits (like Helm, CIT Group etc., and sometimes these rental units are in the original colours of the road that previously owned it, with the roadname marked out and new reporting marks tacked on). There's also plenty of examples of railways directly renting engines from each other. During the 1970s and 1980s Canadian Pacific rented engines from all sorts of other railways to help get through power shortages.
2) Run-through agreements. In some places railroads will have specific agreements to share or pool power in certain areas or on specific trains. Common examples are trains that originate on one railroad and run to a destination on another. Rather than changing out the power when it gets on to the other railway, the engines run all the way through and only the train crew changes.
3) Horsepower Hour (HPH) repayment. When engines are running through in pool service, a lot of accounting goes on to make sure that things are somewhat balanced between railroads. At the end of the day (month, whatever) things need to be balanced up. This usually leads to one railroad loaning an engine (or two or three...) to the other for a specific period of time to balance the books.
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Chris van der Heide
John