This is possible. The most basic configuration that allows TCP/IP connections between your two networks is the following:
- "A" needs to be configured to route between the two networks.
- "B" needs a route telling it that "A" (199.63.31.x) is the gateway to network 10.10.1.0.
- "C" needs a route telling it that "A" (10.10.1.y) is the gateway to network 199.63.31.0.
- You probably need to add
hosts entries on "B" and "C" so they can address each other by hostname. Without this, you must use IP addresses directly.
The exact steps needed will depend on what OS(es) these machines run. For example, Linux distributions put their hosts file at /etc/hosts; Windows machines use C:\Windows\system32\drivers\etc\hosts; Mac OSX uses /private/etc/hosts.
Other considerations you need to take into account are whether "C" needs to be able to access machines other than "B", if you care, or if "C" explicitly needs not to access other machines. You may also want to supply "C" with other services -- access to a DNS server, connections to a Windows Domain, or access to network file shares, for example. These are all possible, but you should know ahead of time exactly what you need to accomplish so you can configure "A" properly.