Application Request Routing question, is this possible?

hi and thank you in advance for taking the time to look at the problem.

here's what we want to do. we would like to present 1 web server to the Internet as the ARR server and have it route all web traffic. the website (domain) ishttp://public.domain.com.  we only want one domain name and everything after that will be virtual directories that can be reached like:

http://public.domain.com/personnel (IIS server1)

http://public.domain.com/payments (IIS server2)

here's the problem we're facing. each section like "personnel" and "payments" are different sections and have their own web servers. Of course each section has more virtual directories but just using these as an example. each web server can be reached internally by it's physical actual address of: 

http://server1/personnel

and

http://server2/payments

so in comes IIS ARR setup. We installed and setup ARR with proxy. created one web farm called public and added both server1 and server2 into the webfarm. thinking that's how ARR works and tested it by typing in

http://public.domain.com/personnel worked but tryinghttp://public.domain.com/payments did not work giving an error of "404 - File or directory not found." it looks as tho the ARR routes to only one server within the server farm and cannot see the secondary server's virtual directories.

to test this we put only server1 into the farm and typing in http://public.domain.com/personnel worked fine but http://public.domain.com/payments did not.

next test remove server1 from the farm and only have server2. typed in http://public.domain.com/personnel did not work but http://public.domain.com/payments worked fine.

you would think if I added both server1 and server2 into a single webfarm that ARR would see all virtual directories in the webfarm and route accordingly. am I missing something?

if ARR cannot do this, what server or software can take two separate webservers with it's on different virtual directories and display as one website domain? for clarification is there a way I can setup 3 separate webservers. each webserver (server1 and server2) host it's own virtual directories but be routed into on web domain? like:

http://server1/personnel

http://server2/payments

and convert them to, with ARR:

http://public.domain.com/personnel

http://public.domain.com/payments