1. All the ship department heads are present on the bridge. At least one is a division head.
Might also be the same thing, this department/division. But preferably not.
3. No one person seems to be over a combined Sciences division. If it was Spock, then he would have reported for the Science Division as Scott did, but he was silent while both Sulu and Piper reported separately.
Or then he's merely economial with self-expression, and feels that only those who arrived are duty-bound to do the song and dance. Navigation would seem to be Mitchell's gig, but he doesn't utter a word, either.
4. Dr. Piper is only over the Life Sciences department. Dr. Dehner is clearly under him, representing the Psychiatry office.
Well, she's also new to the setup; old man Piper would feel compelled to introduce her regardless of the command arrangements.
It almost seems as if Starfleet wanted to send Dehner to spy on the crew - so, way to blow cover here, telling them she's studying their reactions! Especially as she has managed to keep that under wraps until that moment. But perhaps Kirk wouldn't allow random blueskirts to wander on his bridge without a proper excuse? Dehner could well be outside the command structure to better facilitate her role. Kirk's interactions with her are rather indirect, as can be seen later... (He only "asks" her to do things, she does a lot on her own initiative, and when he finally does give her a direct order, she refuses.)
6. Both Helm (Lt. Kelso) and Navigation (Lt. Cmd Mitchell) were under the Engineering Division, perhaps a Ship Systems department under Lt. Cmd Michell which reported to the Engineering Division (Scott)?
Or then those department/division heads just remained silent when their peers spoke.
7. Later in the episode, Kirk calls Spock his "Science Officer", but this title may not be the same thing as the administrative position of Head of the Science Division.
Actually, Kirk doesn't call anybody in particular his Science Officer. Rather, he addresses the entire audience to express his displeasure on a specific subject relating to Science Officers. The subject being, Science Officers must tell the CO the truth, however unpleasant.
Now, there are three people in the room who should pay heed. In a descending order of relevance:
1) Dr. Dehner, who has been remiss in her duties and failed to reveal the full dirt on Gary.
2) Mr. Spock, who now has to step in and do Dehner's job for her.
3) Sulu, a fellow Science Officer who would do well to watch and learn, preempting future mistakes.
Functionally, we can interpret Kirk as chiding Dehner for her omissions, which means Spock can remain a "pure" Executive Officer. Or we can interpret Kirk as defending Spock now that the Science Officer has spoken ill of Gary.
Grammatically, there's no preference there. Interestingly, we can't even tell whether Kirk speaks of his Science Officer or his Science Officers, there being the possessive s at the end of the latter word...
Kirk's next act is to ask Spock to proceed - in interrogating Dehner for further truths. So if Kirk now wants his Science Officer to speak, well, both Spock and Dehner do, but Dehner is the one who does what Kirk really wanted the Science Officer to do.
By the next episode (production order), Spock wears blue maybe indicating that he is now the official head of the Sciences Division.
Perhaps he has even been demoted to Second Officer, for the duration of the earliest episodes (in stardate order) and at least up until "Enemy Within" where he self-addresses himself thus. His duties might not yet encompass the Executive role (although it's difficult to see who else would be doing that one), perhaps excusing certain oddities in what Spock does or fails to do for his captain in these early adventures...
Timo Saloniemi