url method

I thought I posted this earlier today, but it vanished.

I am running Apache 1.3.36 on a solaris 10 x86 box. I purchased a
domain and point it at a dynamic domainname. This is just in case my
ISP changes my IP. My server runs out of my home. eg www.mypurchaseddomain.com
points to mywebsite.homeunix.net which is a DYN DNS name.

I have a client who had a spare windows box and I installed apache on
it. I tried the same setup with his site, but when a user goes to his
site the browser reports the site url as the DYN DNS name and not his
company name. I have the ServerName set in the conf file and both
machines have hostname entries in their etc/hosts file.

What mechanism determines the URL in the browser window and can I
correct this problem?

Thanks

Mike D