What is the proper email etiquette when contacting potential supervisors on short notice? I am currently an MSc student in the UK and am hoping to apply for PhD at the same institution. I have found some potential supervisors but was wondering what is the correct way of wording my email given that I have only about 3 weeks till the application deadline and it's currently the holiday season (end of term). Some things to note: 1) my current field of study is immunology and I'm hoping to apply for a PhD in Medical Sciences, 2) it won't be possible to meet the supervisors in person since I'm back in the States for the holidays.
The first few tips is be cordial and be brief and to the point.

Make sure you use the correct title and spell the name correctly, in fact make sure the entire mail is spell checked and grammatically correct, anything else sends a bad signal.

Begin by stating your purpose for the mail and provide a short but detailed reason for why you approach the person in question. In your case you should also explain the "short notice" and excuse yourself very concisely for the inconvenience this could bring on. 

Follow up by providing a brief statement of why you are interested in the subject but also the school and/or research group. Continue by describing your background and how you see your background being an asset in working in the PhD field.

All this should not be more than 2-3 paragraphs of maybe 5-10 sentences. A short e-mail which manages to say the essentials will be more likely to succeed than a long and very long-winded detailed account (that no-one would take the time to read).

The tone of the mail should be respectful but do not excuse yourself or play "underdog", be confident, brief and to the point. that will impress more.

Finally attach a CV and provide any MS thesis papers you may have.