In English Grammar, before ordinals (first, second, third...), we always use the definite article "the."
Consider the sentence below.
If the same person calls a second time within a 15-minute period, allow it.
In the above sentence, "a" is used before the ordinal "second." As far as I understand it should be "the," not "a." I found this sentence from my android phone.
Please, someone, explain to me why it is "a," not "the?"
"A" is correct here, and "the" would be odd.
Normally, "the" is the article used with an ordinal because what is being described is a specific item in a specific ordering of a set. There is no vagueness about what is being referred to.
In this specific example, however, there is, as yet, no such item to be referenced specifically. The sentence is conditional.
If more than twelve guests arrive, you will need to set up a third table.
In the preceding example, we do not talk about "the" third table because what is primarily being contemplated is a set of two tables. I do not say that "the third table" is wrong, but it would, in American English, be atypical.
In your example, it is not expected that someone will call more than once during an interval of fifteen minutes. If there were such an expectation, it would be proper to say in American English
When someone calls the second time ...
but even then it might better be to say
If someone calls for a second time
if there is no certainty about a second call.
Interesting question.
I haven't found any sources on it, but to me "a/an" with an ordinal can be used when the ordinal is beyond the end of the series expected, or originally counted.
This is especially when there was only expected to be one item (as in your example).
If somebody calls a second time ...
implies that this is unexpected, the norm being only one call.
Similarly
We had arranged to meet twice, but he came back a third time!
suggests that this third visit was a surprise.
Three men were waiting for me. The third carried a briefcase. Then I saw a fourth man in the background.
In all these cases, the is possible, but I think it is less likely in ordinary speech: using it in this context sounds to me like a narrative device to suggest that the speaker (or the hearers) know more than is being said, so that this extra item is not a surprise.
