[susan53]

Remember that "the" indicates shared knowledge - ie something known to both the speaker/writer and listener /reader. Here, this is the first mention of "second time", so it is not yet "known" to the reader. Compare :

I got it wrong at first, so I did it a second time and the second time I got it right.

"I did it a second time" introduces the idea - at this point it's new to the reader. But after that, the concept is "shared" and therefore "the" is used.