Visible crew/equipment: After the Klingon vessel's attack on the Enterprise, while returning fire just as Kirk orders to maintain firing range at 100% dispersal pattern, the set's unfinished wood plank with grey paint drip marks can be seen at the bottom of the screen. Additionally, at the end when Spock goes to stand beside Kirk's chair (also when he walks away) the actor's mark can be seen on the floor by Spock's feet.
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Factual error: When the Enterprise is slingshotting its way back to the "present" it uses the computer to deposit the two men at different times, The Enterprise is traveling backwards in time, so will logically meet the time when the guard was beamed up first, not after Christopher is returned to his cockpit. So the order is wrong. Should be guard then Christopher, not Christopher then guard.
Revealing mistake: "Trek's" new 3rd season uniforms were tighter-fitting than their velour predecessors, presenting a problem for some actors. When the landing party beams down in this episode, the outline of William Shatner's tummy-wrapping girdle is very visibly showing through his costume.
Continuity mistake: After Garth attempts to convince Mr Scott that he is Captain Kirk and to beam him up, while he is banging his fists on the floor in anger he transforms back into himself and the fly like ring on his finger breaks and slides out of shot. It's quite hard to see but it does. The next time we see Garth's hand his ring is intact yet he has not transformed himself again to repair it.