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What is the origin and the meaning of the idiom " on the trot " ?
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It normally means "in succession" or "one after the other without interruption".

For example, "three years on the trot" means three consecutive years.

"trot" means a brisk pace. I suppose the expression originally had something to do with this, but exactly how it came to specifically mean "in succcession" I'm not sure.
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This is actually a Naval term. Submarines always moor alongside on a berth designated as a 'trot'. There are two 'trot' sentries on watch alongside at all times, an 'upper trot' who is responsible for the outside part of the submarine and the gangway and a 'lower trot' who is watchkeeping the internal forward part of the submarine. When there are two or three submarines alongside each other moored alongside they are described as 'two on the trot' or 'three on the trot'.

anonymous

This is actually a Naval term. Submarines always moor alongside on a berth designated as a 'trot'. There are two 'trot' sentries on watch alongside at all times, an 'upper trot' who is responsible for the outside part of the submarine and the gangway and a 'lower trot' who is watchkeeping the internal forward part of the submarine. When there are two or three submarines alongside each other moored alongside they are described as 'two on the trot' or 'three on the trot'.

The inventiveness of the amateur etymologist knows no bounds. It is a testament to the fertility of the human imagination. For every unusual term, there are dozens of such stories, but they remain just fun entertainment until they can be proved. The OED seems to think that this term derives from an expression dating back to at least 1640, when the submarine was a da Vinci drawing.

By the way, it is not in my US vocabulary.