Oooohhh
No slide trombone adept, but I think I recall the octagonal-slides being memorable (which today means 'collectible') - nice you're keeping yours.
If you audition some turn-of(last)-century stuff, Sousa's Band, Herbert L. Clarke and others - Arthur Pryor was the legendary slide-trombone virtuoso, accomplishing what most believed only keyed instruments might. If interested, try to find "Variations on The Bluebells of Scotland" [Arban] - for one selection as will demonstrate his reputation.
Dunno whether his slides were round or octy, but it seems to me hard to swage close tolerances on the inner or outer - all things considered.
The circle represents God.
Ashton