How to locate my Time Machine external drive.

The Stargate drive your using is probably FAT formatted (windows formatted) if it does not show up in Time Machine. To make it compatible with Time Machine, you will need to format it to Macintosh Extended format in Disk Utility. Formatting a drive will erase everything on the drive, so make sure you back up anything that is on your stargate before you format it.

To format your drive, open Disk Utility, click on your drive from the side pane, click the Erase tab, and click on the Format menu, and make sure that Macintosh Extended is selected and not MS-DOS FAT. Then click erase, and your drive should be compatible with Time Machine.

chrisgalz wrote:

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To format your drive, open Disk Utility, click on your drive from the side pane, click the Erase tab, and click on the Format menu, and make sure that Macintosh Extended is selected and not MS-DOS FAT.

Also make sure it has the GUID partition map scheme. SeeTime Machine - Frequently Asked Question #5 for detailed instructions.

See If a disk doesn't appear in the sidebar.

topcop1 wrote:

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Couldn’t unmount disk.

That means something is trying to use it.

But at least it did appear in the sidebar, so that's progress. 🙂

Start from your Install Disc (if you're on Snow Leopard or Leopard), or the Recovery HD (if on Lion), and erase or repair it from there. See the yellow box in Time Machine - Frequently Asked Question #5.

And note that the oldest and latest backup information may not be updated until a backup is completed.

topcop1 wrote:

I have not gone through the recovery stage because today Time Machine stopped showing "Preparing Backup" and it changed to Backing Up. The gigabite indicator kept progressing toward 319Gs for most of the day. then it stopped and went into an hourly backup mode:

Ah, good. 🙂

However, whenever it backs up an icon identical to the Time Machine backup appears labled Time Machine Backups.

That should happen only for backups made over a network. Is your external HD connected directly to your Mac? If so, was it connected to a Time Capsule or Airport Extreme previously?

the clock like icon in the top tool bar does not turn as it previously did whenever a backup was in progress,

Yes, that's new in Mountain Lion. It only spins during the actual copying, not while it's preparing or cleaning-up at the end. We don't know yet if it's a bug or (bad) new design.

Further, the fact that my last backup prior to this was January 4, 2012 indicates that is when I shut down my system and was away for three months. When I returned I hooked up my laptop and it appeared Time Machine was working. But obviously not.

Those backups may have all failed. Hard to tell now, of course. 😟

So, whenever I take my laptop on the road, what is the procedure for disconecting Time Machine? Just turn it off or eject it?

Eject if first. If you don't, the backups can be damaged (the directory structure, etc.).

Aalso, even if it is disconnected, does it still need power?

Nope.

topcop1 wrote:

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It was originally connected through an Airport Extreme and worked fine until recently.

Aha! That explains it.

I hate to have to tell you this, but backing-up that way is unreliable and not supported by Apple. See: Using Time Machine with an Airport Extreme Air Disk. The backups may have been damaged at some time while you were backing-up that way.

Time Machine stores your backups differently when they're done over a network. It puts them in a rather odd container called a sparse bundle disk image, rather than at the top level of the drive as it does when backup are made locally.

If you then connect that directly to your Mac, Time Machine will continue backing-up to it, but some other things won't find it, since they're not looking for a sparse bundle. In those cases, you must manually "mount" the sparse bundle by double-clicking it in a Finder window. See the blue box in Time Machine - Frequently Asked Question #17.

So, I guess it is time to buy a new, and larger external drive.

Maybe. How large, and how old, is it?

It varies widely (see Time Machine - Frequently Asked Question #1), but a "rule of thumb" is, TM needs 2-3 times the space as the data it's backing-up. And has a 3-year waranty.

If your drive is more than about 3 years old, it may be a good idea to get a new one anyway. They don't last forever.

If you want to keep it, your best bet may be to erase it with Disk Utility and let Time Machine start fresh, backing-up to it directly.

Any suggestions for that purchase?

OWC has a good reputation, both for hardware and support.

I recently got one of their Mercury Elite drives: http://eshop.macsales.com/shop/firewire/1394/USB/EliteAL/eSATA_FW800_FW400_USB, as have a number of other users, with good results.

You can also buy enclosures and drives from them separately.

But any decent-quality drive will work well.

A much more convenient alternative would be an Apple Time Capsule. It's more expensive, since it also includes a wireless router (like your Airport Extreme), but was designed for wireless Time Machine backups.