Ian,
Here is a 1926 catalog image of a carb inlet heater, which Douglas called the "Special fuel economy heating pipe". I am sure this is the one Chris calls the "banjo".
Larger viewAlso some pictures of the machined but as yet rough castings. Note the air inlet seems to be a slot on the front face, marked X. There is no communication to the rear face, which is just a heat transfer surface.
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Larger viewA second style is shown below, I think the second type mentoned by Chris. I do not recall a catalog illustration showing this type myself, but it would seem to be for exhaust systems where you do not have the central silencer can directly under the engine, rather a pipe(s) heading to the rear of the bike where it may, or may not, enter a silencer. In this example I would think that (again a rough casting) needs additional machining, to punch a hole or two from the portion that wraps around the pipe into the cored passage leading to the carb. Though why it is not already cast in, it would have helped support the existing core, I have no idea.
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Larger viewI have only seen pictures of these, if I ever saw an actual one, it has escaped my memory.
Douglas kept on with hot air for the carb. Many of the later twenties and thirties models with the carb above the engine, had a inlet tube for the carb that drew air off the rear cylinder. Some models had jacketed manifolds with exhaust heat, or a 'hot spot' in contact with the crankcase to transfer some heat. So it is part of the Douglas tradition! Rather than a hot water bottle, you could lag the inlet manifold with an old tartan kilt.
-Doug