You just need to change <a>to using classes:
<div id="register">
<form id="register">
<input id="first" type="text" /> <a class="first" ></a> <br />
<input id="second" type="text" /> <a class="second" ></a>
</form>
</div>
The jQuery selectors you use are:
$('a.first')
$('a.second')
Search <a>with the class. If you tried to search <a>with an identifier, they should be as follows:
$('a#first')
$('a#second')
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