Brilliant!!! The problem has been isolated to Norton Personal Firewall. When turned off, my LAN set-up works perfectly. It was quite satisfying to see the ping work. I checked everything else. It all works when Norton Personal Firewall is off.
I'm considering leaving it off. Windows Security is running. Norton is on the new computer as a free trial, put there by HP as a promotional offering. But before I do, I'd like to see if I can get the firewall to act properly. I sort of like Norton in a number of ways. I might use it if it doesn't get in the way.
But .... I've been trying to configure the firewall to allow the LAN to operate and nothing I've tried has worked. I've run the network wizard and the networks IP is in the Trusted Zone. Before that, I tried adding individual IP addresses. Network detection is on. I tried fooling with General rules in Advanced configuration just a little bit. But it's a little too advanced for me. I don't know what all those initials stand for.
Temporarily disable all of your security stuff (on the PC's and in the router) - this includes everything that affects your internet connection. Then try the ping - if it works, then your problem lies in the stuff that you've disabled.
Don't forget to turn the protection back on as soon as you can!