I'd like to upgrade the memory on a Northgate (a defunct company)
machine, manufactured by Lan-plus. It has a 2.40 Intel Celeron
processor, supposedly 4 memory slots (though I see only 2), with 256
mb. in one of them. The existing memory stick says this:
P/N 77.10603.11A
S/N: 1104033-9051
256 mb
UNB PC 2100 CL2
Apacer
It has 8 chips on it.

wilsonqly-ga.. This is what I found on your ram. 
http://www.apacer.com/apacer_english/product_html/generic_memory.asp

PC2100
184pin DIMM, 184pin Reg. DIMM, 200pin SODIMM
128MB/256MB/512MB/1GB/2GB
x64bit, x72bit, TSOP, BGA
2.5V CL2, CL2.5

So it is DDR PC2100, does this look familiar?

Crucial Memory System Scanner

 

Thinking of upgrading your RAM? 

The Crucial System Scanner automatically detects your computer's
information and gives you the details you need to purchase a
compatible upgrade. To run the Crucial System Scanner, you need
Microsoft Windows 95 or higher.

The Crucial System Scanner will load a small tool (appropriately 77KB,
which will take about 15 seconds to download on a dial-up connection)
on your computer that will scan your computer's hardware, then show
you a page with important information that will help you to select the
right memory upgrade for your computer.

http://www.crucial.com/systemscanner/

Many thanks for the DDR PC2100 guidance, Bumble. I'll reopen the
machine & take a look. Lri, thanks, but I tried the Crucial scanner a
while back & it told me I was out of luck; they had nothing
compatible. (Maybe I should've given up at that point.)

DDR RAM is backwards compatible, this means that if you find RAM that
is the DDR 2100, or better it will work, I know that the office stores
are still selling DDR 2700 fairly cheap, and this should work in your
system.  Always remember to put the largest stick of RAM in the first
lot, meaning if you have the 256 in the primary slot and you buy a 512
replace the 256 with 512 and then move the 256 to the secondary slot
this will improve performance.

Don?t be afraid to put your hands in the machine just watch for the
moving parts and unplug it first.