Perigosto stick

A perigosto stick was an object which looked like a thin, silver probe with a tiny sphere on top. (PROSE: Love and War) Students at the Time Lord Academy used them to play games of four-dimensional juggling. (PROSE: Seeing I) The Doctor also claimed that it was a musical instrument. (PROSE: Deadly Reunion, AUDIO: Voyage to Venus)

Gallifrey
The Doctor was in his early nineties before he could bring himself to admit that he still liked to play with his perigosto stick. The other students teased him mercilessly about that for about fifty years. (PROSE: Beltempest)

While in the academy, the First Doctor conducted an experiment involving a perigosto stick and a temporal feedback loop which resulted in High Tutor Albrecht's regeneration. For his actions, the Doctor was appeared before the Academy Board and was reprimanded by Borusa, the head of Prydon Academy. (PROSE: "Academic Progress")

The First Doctor and his friend Ruath once electrified Cardinal Borusa's perigosto stick. (PROSE: Goth Opera)

He played lead perigosto stick in Gallifrey Academy Hot Five, while the Master was on drums. (PROSE: Deadly Reunion)

On an occasion when believed he had killed the Third Doctor, he ruefully thought back to a youthful prank when he and the Doctor had put glue on the Lord President's perigosto stick. (PROSE: Doctor Who and the Dæmons)

The Third Doctor once asked Sergeant Benton to pass him a perigosto stick from his lab bench. (PROSE: The Devil Goblins from Neptune)

Venus
The Third Doctor once told a humorous story in which the punchline was, "Never trust a Venusian shanghorn with your perigosto stick." (TV: The Green Death) The Thraskins used the perigosto stick to put the Shanghorns to sleep. (AUDIO: Voyage to Venus)

Behind the scenes

 * The perigosto stick was the basis of a minor running joke by Doctor Who novelists. Building upon the minor mentions in The Green Death and Doctor Who and the Dæmons, novelists would make elaborate mention of perigosto sticks without explaining all that much about what they actually were or might have been used for.