Doesn't the ethernet router already provide internet access for both machines? It sounds like you may be sharing something that doesn't need sharing.
I'd suggest disable the internet sharing, reboot both Macs & see if they can both connect to the internet. I'd also disable wifi so you can see if ethernet is working correctly.
Ethernet sharing is only required when you have an internet connection that doesn't natively support ethernet or wifi, like tethering to a phone or a USB only modem.
Is the modem 'USB only' or does it have ethernet?
Internet sharing also only works from one network port to another, for example I can share the internet from my ethernet to other devices over wifi, but not over ethernet. Since the iMacs only have one ethernet port I suspect you are not sharing over ethernet, wifi will be involved somewhere along the line, unless you have a USB/ Thunderbolt ethernet adapter?
Thanks for your thoughtful reply, and I apologize for my delay in responding.
You've solved my problem, albeit indirectly: I turned off ethernet sharing, and logged my older iMac into the newer iMac's Airport Wifi network. (You had suggested that I had been in fact using Wifi instead of ethernet, but the older Mac's network panel had clearly shown that no wifi network was connected).
The older machine's internet performance using Wifi isn't sizzling, but it's adequate. To give you an idea how bad ethernet sharing was working. I have Eudora, a venerable email client, installed on the older iMac, and internet was was so slow that Eudora sessions timed out 19 times out of 20. (Yes, I should migrate Eudora to the newer machine, but that's part of a larger, tricky migration problem that needs more time than I have right now.)
I'd like to pursue an important issue you raised. You asked: "Doesn't the ethernet router already provide internet access for both machines?"
The answer is "no," because I unfortunately have no idea how to configure an ethernet router; I have configured several LinkSys Wifi routers, but only by filling in numbers by rote supplied by tech support. And I guess I've set up a Time Capsule that supports multiple ethernet connections, but I didn't have to figure it out; the software configured itself.
So, can you point to a primer that would teach me how to configure an ethernet router for sharing the internet supplied by a cable modem among a small number of Macs? I actually did do some google searches before coming up with the idea of using the built-in Internet Sharing via ethernet, but didn't find anything.
Message was edited by: tttny
Since you have one working Ethernet connection, get a switch, and then run 2 Ethernet cords from the switch - one to each computer. Saves configuring and is easy. I have several switches in the house and on 1 of them, 3 Time Capsules, a printer, and a computer all run off the same switch. Here is an example, although you might be able to find them cheaper elsewhere.
Are you able to find the Linksys router model number on the case or in the admin control panel? We may be able to point to help documents for you.
My guess is that if it has multiple ethernet ports they will probably be configured by default to work as a networking switch (like Eric mentioned). So it should be a case of plugging in the Mac via ethernet, making sure that it gets an IP address on the ethernet connection. If the Mac fails to get an IP address it may need additional configuration, but it should work if the ethernet port is set to use DHCP.
The router is almost certaintly setup to hand out addresses via DHCP, so once it sees the Mac it should assign an address & things should work.
It's an Asante router that I picked up years ago. It doesn't have a WAN port, which I remember puzzled me, . . . and looking at it more closely I see that it's a hub, not a router. But yes, it just dawned on me that I could check the Asante website (duh!).
In retrospect, the interesting thing is that internet sharing via ethernet worked at all . . .
Drew and Eric, thanks for your help.