AJAX call over a https service doesn't work

I'm developing a node.js REST service which is being used by a web client with javascript. I was currently offering this services over HTTP. Now I have changed the server to offer them over HTTPS and ajax calls have stopped working.

When I set up the server to be over http this client code works (shows the "success!" alert):

<html>
<head>
    <title>Client</title>
    <script src="http://code.jquery.com/jquery-1.8.0.min.js"></script>
    <script type="text/javascript">
        $(function () {
            $("#go").click(function (e) {
                $.ajax({
                    type: "GET", 
                    url: "http://localhost:3010/users", 
                    data: { username: "test", password: "12345" },
                    success: function(data) {
                        alert("success!");
                   }
                }).error(function (data) {
                    alert("error");
                });
                e.preventDefault();
            });
        });
    </script>
</head>
<body>
    <a id="go" href="#">go!</a>
</body>
</html>

When I change the server to use HTTPS and change the url in the client code:

$.ajax({
   type: "GET", 
   url: "http://localhost:3010/users",
   ...

When I use CURL to test the server with the https calls it does work perfectly:

curl --insecure "https://localhot:3010/users?username=test&password=12345"

I don't understand why the ajax calls fail when using the https services.