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Here the "second" means 2nd, not 1/60 minutes.

I'd appreciate it if someone gives examples and detailed explanations.

Sorry for lack of context.

Here's an example:

"Ottawa police have apparently discovered a second suspicious package at the Canada Post sorting facility in Ottawa. The Ottawa Citizen’s Meghan Hurley is reporting that package contained a hand."

Why do they use "a second package", not "the second package"?

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    Can you give some more context? Like some example usages? I can tell someone something for the second time, I can have a second helping of potatoes, Lady Gaga can win a second Grammy award, a basketball game can resume in the second half, etc. Are you sure this isn't just a question about the difference between "a" and "the"? – J.R. Feb 10 '15 at 23:33
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    This is more than the second time I have written such a comment, but I will give you a second chance. Please, everyone… details. Please – user3169 Feb 10 '15 at 23:39
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This answer is going to assume that your question ultimately expands to be:

How do I tell the difference between the ordinal and the unit of time?

Answer: context.

Both the ordinal and the unit of time can be referred to using either article (the definite the or the indefinite a), so that's not going to help.

Commonly, when referring to the ordinal second, you'll see it describing something else:

  • The second car has crossed the finished line.
  • A second racer has crashed into the wall!
  • The competitor must score a second perfect to advance...
  • This is now the second time that the racer has earned second place.

Adding to the confusion, the ordinal can refer to "prize" or "place" without adding those additional words and still be valid English:

  • Prize
    • The second prize goes to Nutley.
    • Second goes to Nutley.
  • Place
    • James lost to King, earning a second place finish.
    • James lost to King, earning second.

However, you'll commonly see ordinals with other ordinals:

  • I had no idea the car could have leapt from fifth to second!
  • That puts Mary in first, Jesse in second, and King in third...
  • First, we break into the bank. Second, we loot the bank. Third, we all get arrested...

When second is being used as a unit of time, they'll have a context relating to time:

  • I'll be with you in a second.
  • I'll fire the second I see it move.
  • The second the item goes on sale, I'm buying it.
  • A second isn't much time to react.

Just to reiterate, the context is what will tell you one way or the other, and why writing grammatically correct English can be very important. Take the following terse sentence, do you know what they mean?

That's the second second! Wait, do we have a second second too?

  • Is the speaker talking about about two seconds of time?
  • Is the speaker incredulous at a competitor earning two second-place finishes?
  • Is the speaker confused about there being more than one person earning second-place?

Without that context you just don't know!

  • And, just for fun, it might refer to a duel, where one of the duelists shows up with two seconds. – WhatRoughBeast Feb 12 '15 at 1:20
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"A" or "an" is the indefinite article in English and can be thought of as meaning "one." On the other hand, "the" is the definite article.

The definite article is used to denote persons or things that are familiar to the listener or reader, such as things that are under discussion or which have already been mentioned.

"The second suspicious package" would not be appropriate in your example unless the suspicious packages were already familiar to the reader, such as if they'd been mentioned earlier in the news article, or if it were widely known that a number of suspicious packages were being sought.

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