load more posts by category

You will need to load the next 10 posts with an AJAX request and keep the current offset stored in the DOM.

Attach an ID to the <span> element which is your button, in my example I have used #load_posts.

Then using jQuery, you can easily setup the click event for the button to fire the AJAX request.

So the button:

   <span class="ar-bottom-link-tabs" id="load_posts">
        <span>
            <a href="#">Load more &raquo;</a>
        </span>
   </span>

And then add the jQuery underneath outside of the loop:

<script type="text/javascript">

    // We have already loaded 10 posts, so set it up as 10 by default
    var currentOffset = 10;
    var categoryId = 8;

    $('#load_posts').bind('click', function(e){

        // Stop the link reloading the page 
        e.preventDefault();

        // Asign variables to an array
        var data = {
            offset: currentOffset,
            catid: categoryId
        };

        // AJAX $_POST request to PHP file
        $.post('path_to_php_file.php', data, function(response) {

            // do something with the response... loop through and append to the tab content
            // so you could return the output as JSON

            // Add 10 to the offset
            currentOffset+10;

        });         

    });

</script>

Then in the PHP file used for the AJAX request just repeat your get_posts() call:

$args = array(
    'cat' => $_REQUEST['catid'],
    'offset' => $_REQUEST['offset'],
    'showposts' => 10
);

$posts = get_posts($args);

I will let you fill in the gaps but this should send you on your way in the right direction.