You've got a couple of options here:
Save the variables in a cookie in success.php and then read them in completed.php.
Save the variables in a file/database and then read them again.
Pass them on the URL on the redirect, eg page.php?var1=blah&var2=blahblah
You could also play around with a form on success.php that had hidden fields, use javascript to fill them in and then submit() the form - although this then becomes a javascript issue, not php one.
Good luck!