While researching our ancestors we often don't find a female descendant because she changes her name when she gets married. That is particularly true when there are many young women in the same town with the same name, and the parents are not listed on the marriage records. We all have a few of those. Immigrants may have had their name changed by agents who couldn't pronounce or spell their names as they came through customs.
But sometimes we can't find a person because, for whatever reason, they change their name. An ancestor may decide to change his name for nefarious reasons, in which case we may never find them. But to legally change their name, a person submits a petition to the court and places an ad in the local paper.
I can see why this couple decided to change their surname to Fox. I guess he thought if he was changing his name he may as well go all out!
In some places if a woman gets divorced and wants to revert to her maiden name she must also petition the courts, as Mrs. Baer did in 1934.
In the UK notices are posted in the Gazette. Type "name change" in the search box, then in the left hand column check "people".
There are many reasons a person will change his/her name. My husband's aunt, Marie Otilia Nolin had her name changed to Sister Thérèse Joséphine when she became a nun. So religious orders are another place to look for an elusive ancestor.
Listed below are some links to people who have, for some reason, changed their name.
List of Persons whose names have been changed in Massachusetts - 1780-1883
How to change your name by Deed Poll - UK
English Province Society of Jesus - alphabetical catalogue of members who assumed aliases or by-names, together with said aliases. - 1875
Handbook of fictitious names: 1868
Register of changes of names during the war by deed poll and enrolment as announced in the "London Gazette." 1919