Diana Muldaur, who plays
Dr. Ann Mulhall, would later play Dr. Miranda Jones in "Is There In
Truth No Beauty?" as well as Dr. Kathryn Pulsaki in the second
season of "Star Trek: The Next Generation".
Anne Mulhall is a doctor in astro-biology, but is wearing red instead of a science/medical blue uniform.
Kirk at first opposes Spock beaming down with him. After two seasons of always beaming the three most senior officers into any situation -- known, unknown or actively hostile -- it seems odd that here he's concerned with leaving the ship without its two senior officers.
Where are Kirk and Mulhall's consciousnesses when Sargon and Thalassa have their last kiss? Previously it was established two consciousnesses couldn't share a single body, but there's nowhere to "store" Kirk's and Mulhall's. Apparently they're both back in the main computer but this isn't very clear.