/wordpress slug on primary site

Hi.
I just installed multisite and I can’t quite seem to understand how the url is defined properly.
When multisite is enabled my primary site’s slug have /wordpress after the domain name.
eg: http://www.domain.com/wordpress/sample-page
But the sub domain doesn’t have /wordpress. – Either does it’s wp-admin.

When single site i have this setup:

$localurl = 'local.localurl.dk';
$url = PROTOCOL . $localurl;
define('WP_HOME', $url);
define('WP_SITEURL', $url.'/wordpress');
define('WP_CONTENT_URL', $url.'/content');
define( 'WP_CONTENT_DIR', $_SERVER['DOCUMENT_ROOT'] . '/content' );

And it works perfectly.
But with multisite I have these extra definitions:

define('WP_ALLOW_MULTISITE', true);
define('MULTISITE', true);
define('SUBDOMAIN_INSTALL', true);
define('DOMAIN_CURRENT_SITE', $localurl);
define('PATH_CURRENT_SITE', '/');
define('SITE_ID_CURRENT_SITE', 1);
define('BLOG_ID_CURRENT_SITE', 1);

It’s a bit difficult to explain.
Hope you can help.

Thanks.
Regards.
Jimmi