The Palace of St. James was where the royals like King George III lived prior to Buckingham Palace. They would host balls in celebrations of "official nativities of the King and Queen" (Thompson) where people of Sir William's rank and higher attended. Sir William is referencing an earlier scene in chapter six asking Darcy if he ever danced at St. James's. (http://www.jasna.org/persuasions/on-line/vol33no1/thompson.html) (https://janeaustenslondon.files.wordpress.com/2013/04/st-jamess-palace-1-copy.jpg)

"That wherein one has fallen behind. A duty or liability overdue and still remaining undischarged, esp. a debt remaining unpaid." (OED).

"The land of lakes; spec. the region of the English lakes, consisting of parts of Cumbria and Lancashire" (OED).

"The Lake country is in the far northwest of England" (https://www.pemberley.com/janeinfo/ppjalmap.html#longbourn))

In 2019, 300 pounds would be equivalent to $30,372.96 while 400 pounds would be equivalent to $40,497.28. 1802 is the closest year to input the information because in the following paragraph, there is an indirect reference to the Treaty of Amiens which took place between 1802-1803. https://www.uwyo.edu/numimage/currency.htm

"To settle definitely" (OED).

This was in reference to the peace Treaty of Amiens in 1802:

"Treaty of Amiens, (March 27, 1802), an agreement signed at Amiens, Fr., by Britain, France, Spain, and the Batavian Republic (the Netherlands), achieving a peace in Europe for 14 months during the Napoleonic Wars." (https://www.britannica.com/event/Treaty-of-Amiens-1802).

"To declare with confidence, to justify" (Johnson).

"Diligently, continually" (Johnson).

"The act of proving true; confirmation, attestation, proof; the action of formally or authoritatively declaring good or true; sanction" (OED).

"To happen to; to become of" (Johnson).

"Before this time, before now, in time past, formerly" (OED).