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Your first step towards triggering a leadership challenge is to get 15% of Tory MPs to sign a letter to the chairman of the 1922 Backbench Committee2 signalling your lack of confidence in the leader. The 1922 Committee will then call for a vote of no confidence in the leader.
If over half the incumbent Tory MPs vote against the current leader, s/he is forced to stand down and an election takes place to choose a new leader.
A New Leader
Of course, once you've got rid of your leader, you now have to find a new one and this starts with MPs announcing whether they intend to stand for the leadership. If more than one MP announces their intention to become leader, a contest is held. All Conservative MPs vote on which person they would like to become leader and at the end of the ballot the person with the least votes is eliminated3. A new ballot is held and another person is eliminated. This continues until only two are left. Every paid-up member of the Tory Party is then allowed to vote for the candidate of their choice - and the MP that receives the highest number of votes becomes leader.
Of course, leaders of the Tory Party have been known to leave after General Election defeats or just when they've decided enough is enough. So don't give up hope if you aren't be able to get rid of your leader this time round; it'll happen eventually.
History of the Leadership Challenge and Elections
There have been many reforms of the Tory leadership elections and ways of challenging the leader since 1965, when the leadership ballot was opened to all MPs4. Here is a short history, both of the reforms, and memorable moments:
1965 - 'Magic Circle' abolished. The ballot was opened to all Tory Members of Parliament and Edward Heath won.
1989 - Sir Anthony Meyer became the first MP in history to challenge the leadership of a Conservative Prime Minister, Margaret Thatcher. Not that he was successful, he got just 33 votes in the vote of no confidence6.
1990 - Michael Heseltine successfully challenged Mrs Thatcher's leadership. Well, sort of. She actually won the vote, but Cabinet wouldn't back her, so she stepped down. John Major beat Heseltine in the ensuing leadership contest.
1995 - John Major took the odd step of resigning as PM and standing for the leadership again. He did this in order to make a point to the less loyal members of the party, who were beginning to grumble about his leadership. He won, but went on to lose the next General Election.
1998 - William Hague reformed the election, making the last round open to all members of the Tory Party. The new system was first used in 2001, when Iain Duncan-Smith won against Ken Clarke.