"the judge's seat in wig and gown"

This is a reference to the traditional attire for a British judge, whose wig is made of horsehair.

According to the Sunday Times (13 July 2007), "Judges are to end centuries of tradition and abandon the wearing of wigs and gowns in hundreds of civil and family cases. The decision to abolish the 300-year-old horsehair headgear, along with wing collars and bands, was announced yesterday by the Lord Chief Justice, Lord Phillips of Worth Matravers. However, in a compromise ending one of the most hotly disputed legal debates of recent years, judges sitting in criminal courts will continue to wear their wigs."

Attire for various judges around the world can be seen here.

The video below, which shows traditional British courtroom garb, is from another work by Agatha Christie, Witness for the Prosecution.