I'm interested in taking up archery. Attending the closest club would require that I take the train there as I cannot drive.
Would I be able to take a bow and arrows, secured in a case, on the train?
The train operator is Northern Rail in the UK.
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If the train operator allows it, yes, you would be permitted to do so.
There are no laws specifically against the carrying of bows and/or arrows, nor is a licence required for their purchase.
Bows can be carried openly with less issue, but arrows (because of their ease of use as a weapon) should be stored securely and safely, because
this protects you and others from accidental injury;
it prevents others using the arrow as a weapon;
it is affirmative evidence for a defence of lawful purpose, if you are accused of carrying an offensive weapon.
The latter is the greatest risk you may face from a legal perspective, so you should follow best practise similarly to firearms and ammunition: store and transport the bow and arrows safely, securely, separately.
In addition to the answer by @Nij, this would be like people taking potentially offensive weapons to and from martial arts events. I was taught that any offensive weapons should be carried in a bag, at the bottom of a separate bag. This was to make it hard (and shown to be hard) to access. I would do a similar thing with your arrows.
From my limited understanding, an offensive weapon is only such when it can be used, or looks like it can be used (e.g., replica weapons).