The Paillard family was widely involved over many years in music box making in Sainte-Croix in various firms involving combinations of clan members related by blood or marriage. Paillard-Vaucher & Fils, who made your box, was founded by Auguste Paillard-Vaucher in the 1830s. The “fils” (son) was Arthur Paillard who succeeded his father Auguste in the business.
During the heyday of music boxes, Britain and America were important markets for Swiss music box makers, thus English language usage on the boxes. Many of them had agents or sales rooms in Britain and America.
If you get really curious about the industry, especially as it centered around Sainte-Croix, an excellent book to read is “The Music Box Makers: The History of Music Box in Sainte-Croix,” by Jean-Claude Piguet. For an overall view, one of the very best is “The Musical Box: A Guide for Collectors” by Arthur W.J. G. Ord-Hume. You will find a complete copy of your box’s tune sheet on the lower-left corner of page 323 of the latter. You are fortunate that the majority of yours is still there, as tune sheets are often missing and some 100+ year-old melodies are now hard to identify. The card stock used to make them often becomes brittle with age and they just disintegrate.
If you were a person who did not play an instrument and wanted music in your home before the phonograph was invented a good way to do it was buy a music box and many thousands of people did.
In any event, I hope you enjoy your music box, and who knows maybe you too will catch the music box collecting bug. Just don't be like me and collect bunches of things. It can get expensive and very crowded!
Clay