In Dumbledore’s day (although not in Armando Dippet’s), the office contained a number of spindly-legged tables supporting various frail “curious silver instruments” (CS12), the function of which Harry did not know. They usually whirred and emitted little puffs of smoke.
In Order of the Phoenix (OP22), the instruments were still when Harry went to see Dumbledore after dreaming of the attack on Arthur. Dumbledore activated one of them with a tap of his wand, upon which it emitted green smoke. It showed the snake and then split in two, after Dumbledore posed the question "But in essence divided?". Dumbledore then tapped it again with his wand, and silenced it.
The silver instruments were broken when Dumbledore struck down Fudge, Dawlish, Shacklebolt and Umbridge later in Order of the Phoenix (OP27). But they had "repaired themselves" when Harry visited the office again at the end of Order of the Phoenix (OP37), "puffing and whirring serenely" once more. One of them was then destroyed by Harry, who threw it against the wall in his rage. He threw the table afterwards.
In Half-Blood Prince, the silver instruments were once again restored, puffing and whirring, as were the tables upon which they stood (HBP10).
In PS/SS5, Harry passed a shop in Diagon Alley which sold "strange silver instruments". Would those be the same instruments Dumbledore had? Also, in Order of the Phoenix, they found a silver instrument in the drawing room, which tried to pinch Harry's arm (OP6). However, as it already looked unpleasant, it had most likely been bought in Knockturn Alley.
The "in essence divided" enigma eluded fans for years. After the release of Deathly Hallows, Rowling discussed it in the Bloomsbury Chat on July 30, 2007 and the interview on Pottercast on December 17, 2007. However, in each of those two interviews she took a slightly different approach to the matter:
- Bloomsbury chat: it meant that the snake's essence was divided, it shared part of it's soul with Voldemort. As Harry was "the unintended horcrux", he was able to look through the snake's eyes. Rowling mentioned that Dumbledore was helped by the Pensieve in reaching this conclusion, which is obviously an incorrect statement.
- Pottercast interview: the snake is the symbol for Voldemort, who is "in essence divided", having created a horcrux. He had split his soul.