Goodness, Jane WATERSTON's death was Sept 19, 1859. So baby was born abt Sept 10, 1859. Informant signs as Leonard WATERSON. Can't be same William as born Nov 1857 to Leonard and Jane Beattie. So lets find a baby William born Sept 1859.
I found a William Waterston born 1859 but it was in March and his parents were John Waterston and Margaret Corvan.
Ah, here it is. Joseph Waterson, Sept 8, 1859, 7 High Street, Leonard Waterson, lemonade Salesman, Jane Beattie.
On the marriage record, Jane's parents given as William Beattie and ?Jannet (can't read it) Ramsay.