a second time表示又壹次、再壹次的意思，相當於another time。而the second time表示第二次，前面出現了第壹次，表示順序的。
a second time例句：
The gun was fired and Beaton was wounded a second time
槍響了，比頓再次受傷。
He began counting them and then, with growing perplexity, counted them a second time.
他開始數，然後更加困惑，於是又數了壹遍。
The Masters golf tournament was won a second time by the American Ben Hogan.
美國的本·霍根再次奪得高爾夫大師錦標賽冠軍。
And now I've lost him a second time, and this time for ever!
現在我又壹次失去了他，這次是永遠失去他了！
When I read it for a second time, I had such a feeling.
我讀第二遍的時候，我發現了這樣的心。
the second time例句：
He was married for the second time, this time to a Belgian
他第二次結婚，這壹次是和壹個比利時人。
That is the second time you have made that observation.
那是妳第二次作那樣的申述了。
It's only the second time a potential heir to the throne has married a commoner.
壹位可能的王位繼承人與平民締結婚約，這種事還只是第二次。
For just the second time a look of emotion creases his face.
他的臉上，第二次流露出了感情。
The second time he fired his gun jammed
他開第二槍時槍卡殼了。