First, I suppose I should explain a “by-election” for those of us in the U.S. The UK Parliament page does a wonderful job of summarizing the key tenets of the situation.

1. A by-election occurs when a seat in the House of Commons becomes vacant between what we call “general elections.”
2. A by-election can become necessary if a Member of Parliament (MP) dies, resigns, takes a seat in the House of Lords, is convicted of a serious crime, or declares bankruptcy. [Please recall, especially up through Queen Victoria’s reign, we would find the title holder for a peerage in the House of Lords, while the man’s eldest son and heir sat in the House of Commons.]
3. If the MP simply changes his political party, a new election is not always called.
4. With an open seat in the Commons, customarily, a Member of Parliament from a neighboring constituency covers both seats until a new member is elected.
5. The Chief Whip of the political party the former MP claimed starts the process of a by-election to fill the seat. [In the U.S., this would be either the Speaker of the House or the President Pro Tempore of the Senate.]
6. “Whips are MPs or members of the House of Lords appointed by each party to inform and organise their own members in Parliament. One of their responsibilities is to make sure that their members vote in divisions, and vote in line with party policy. It is the party whips, along with the Leader and Shadow Leaders of each House, that negotiate behind the scenes to arrange the day to day business in Parliament – a process often referred to as ‘the usual channels’.
7. “The Chief Whip of a party is a senior position and they are usually involved in top-level party discussions.” [UK Parliament]
8. The Chief Whip begins the process by “moving the Writ,” a motion requesting “that the Speaker do issue his Warrant to the Clerk of the Crown to make out a new Writ for the electing of a Member to serve in this present Parliament for the constituency of …. in the room of…”
9. The Speaker puts the question to MPs to decide whether to agree to the motion.
10. If MPs agree it becomes an Order for the Speaker. The Speaker then issues a Warrant to the Clerk of the Crown who then sends the writ to the Returning Officer. The timetable is between three and four weeks of the writ being issued, but such is not a rule of thumb.
11. Generally, it takes three months for the process to know fruition. It can, however, take longer. Procedure wise, if the situation occurs toward the end of a governmental session, a seat may be left vacant until a new Parliament is convened.
12. If more than one vacancy occurs, the elections tend to coordinated to fall on the same date.

Here are some of the other things I know of such an election . . .

Serious practical difficulties also arise in the calculation of the size of a `voterate’ from polling figures alone, because of the nature of elections in double-Member constituencies. Even in a straightforward contest between two pairs of candidates, not all freeholders would vote a ‘straight’ party ticket. Some cross-voting, or ‘split’-voting, would be inevitable, even in constituencies polarized by party interests. 

Pollbook analysis for the counties of Buckinghamshire and Westmorland, for example, in the years after 1701, shows an already low percentage of ‘split’ votes declining at every election, but never disappearing completely. Other voters would ‘plump’ for their favoured candidate by casting a single vote. Calculations based on the votes cast for each candidate thus offer no more than a rough guide to the total number of freeholders polled. 

The method adopted in the constituency articles has been to add up all the votes cast and divide by the number of seats being contested, which produces a minimum figure. This obviously works best when four candidates have contested two seats. Three-cornered contests, and those rare occasions in which there were more than four candidates, make the arithmetic more problematic and the results even more approximate. 

Each borough elected two representatives to the House of Commons, and, in a general election, each elector had TWO votes in filling a borough’s Member of Parliament seat. Remember most boroughs had two seats in the Commons. Those votes could be split or both placed for the same candidate.

Yep, two votes, (Oh, what we Americans could do with such manipulations!!! LOL) with which voters could vote straight (both votes to one party), split (one each for candidates of two different parties), or vote “plump” (cast only one vote for one candidate; the other vote goes unused.). I am thinking the by-election would give voters only one vote, but I cannot find that in my notes. I am hoping I did not make that up.

These are double member constituencies. Or like in London where 4 voters had to fill 4 seats. I think there were candidates for each seat, and voters could vote for someone for each vacancy. I don’t know if this method changed before 1832. A by-election concerned one seat so would involve one vote (I hope.)

By-elections, when fought between two candidates over one seat, produce a precise aggregate of votes cast, but are by definition exceptional, their circumstances (with not every major interest in a county necessarily involved) proved to be conducive to a low turnout.

Other Resources:

By-elections, Institute of Government (very thorough overview of the process)

“Constituencies and Elections,” The History of Parliament

UK Parliamentary By-Elections

UK Parliamentary By-Elections in Great Britain