Depending on the domain names in the applications' URLs the second application may be able to read the contents of the cookie set by the first application.
If the first application runs at
http://nonprofit.org/a/ and the second application runs at
http://nonprofit.org/b/ then the second application will get the cookie data because the browser will send it via HTTP headers.
If the applications are on different subdomains like
a.nonprofit.org and
b.nonprofit.org then you may have to update the code that creates the cookie to specify the cookie's domain be
nonprofit.org. That will tell the browser to send the cookie data to both applications. Of course then
c.nonprofit.org will also get sent a copy of the cookie data when the browser visits it and that may not be desirable for privacy reasons.
Update: ++
Your Mother is right about the path. For some more cookie information visit
http://www.cookiecentral.com/faq/#3.2