Reference

According to the search results, specifically from the Wikipedia page on "List of Hertfordshire boundary changes" and other sources, the districts of Middlesex that were transferred to Hertfordshire by the London Government Act 1963 include:

1. Potters Bar Urban District: This district was transferred from Middlesex to Hertfordshire. Source
2. Barnet Urban District and East Barnet Urban District: These districts were also transferred to Hertfordshire. Source

These changes were part of the reorganization of local government boundaries as defined by the London Government Act 1963. The image below illustrates the changes in the London area, including the districts mentioned:

This visual provides a historical context of how the districts were restructured around the time of the Act.