It is possible that the silver instrument was able to provide magical confirmation of facts:
Dumbledore now swooped down upon one of the fragile silver instruments whose function Harry had never known, carried it over to his desk, sat down facing them again and tapped it gently with the tip of his wand.
The instrument tinkled into life at once with rhythmic clinking noises. Tiny puffs of pale green smoke issued from the minuscule silver tube at the top. Dumbledore watched the smoke closely, his brow furrowed. After a few seconds, the tiny puffs became a steady stream of smoke that thickened and coiled in the air ... a serpent’s head grew out of the end of it, opening its mouth wide. Harry wondered whether the instrument was confirming his story
Order of the Phoenix - page 415 - British Hardcover
It's possible the color of the smoke is significant; obviously the shape of two snakes is hugely significant. The two combined together could be hinting at Slytherin House and, ultimately, Tom Riddle/Voldemort, the Heir of Slytherin. I think this would be entirely incidental, though. What the two divided snakes represented -- the essence divided -- was the fact that Harry had a part of Voldemort's soul inside him, and Dumbledore began to understand the enormity of it.
Rosi: What does 'in essence divided' mean?
J.K. Rowling: Dumbledore suspected that the snake’s essence was divided – that it contained part of Voldemort’s soul, and that was why it was so very adept at doing [Voldemort's] bidding. This also explained why Harry, the last and unintended Horcrux, could see so clearly through the snake’s eyes, just as he regularly sees through Voldemort’s. Dumbledore is thinking aloud here, edging towards the truth with the help of the Pensieve.
J.K. Rowling Web Chat - The Leaky Cauldron - 07.30.07
I believe the above paragraph refers to both Nagini's status as a Horcrux and Harry's connection to Nagini and Voldemort. I also interpret the last sentence to mean Dumbledore stored his memory of the "essence divided" silver instrument and reviewed it (I'm guessing numerous times) in the Penseive.