Hi, Patrick.
I've had a similar problem, due to a computer that stubbornly wouldn't install properly from CD. I used the image found on Mandrake CD 1 called ./images/network.img. I assume that Slack has something similar. The next step is to get that image not just copied, but written 1:1 on the boot floppy. To do that, login as root (or su to root), change to the directory with that image file (network.img in this example) and type "dd if=network.img of=/dev/fd0". (assuming that your floppy drive is /dev/fd0)
This should make a boot floppy that will tell your computer to start looking for Suse installation files on the network. You'll probably have to specify the name or ip address of the server that has them (your other computer).
It is highly likely at this point that you'll also need an "additional drivers floppy". Again, on the computer that already has linux installed, go back to the directory that you got the "network.img" file from and look for something to the effect of "network_drivers.img". Use the same process as above to transfer the image to a DIFFERENT floppy disk (unless you don't think the installation will need your boot floppy...but I wouldn't count on it
). (The command you need to use is "dd if=network_drivers.img of=/dev/fd0".)
By this point, you should have smooth sailing. One point that came up while I was doing the install this way was that when I needed to change my cd on the "source" computer, the "target" computer assumed that the packages/images/whatever were bad and the install consequently crashed. Don't let this happen to you! Be sure to copy ALL of the CDs into the directory you're going to be sharing over the network for the "target" computer to install from.
To make this easy, you can do something to the effect of inserting CD 1 into the drive and typing "cp -Rvf /mnt/cdrom /your/shared/directory". Replace "/mnt/cdrom" with wherever your computer mounts your CD to, and replace "your/shared/directory" with the path to whichever directory you're letting the other computer install from. Now just insert the next CD, and assuming they don't use any of the same filenames in the same places, you can just repeat the "cp ..." command. If you don't know if there are redundancies in the file names and directory trees, just remove the "f" option from the command, so that it looks like: "cp -Rv /mnt/cdrom /your/shared/directory". This way it will prompt you for overwrite if it looks like something is about to get copied over.
I hope this works for you. If not, you can probably use most of the steps involving copying the files into the shared directory, and perhaps simply finding a more appropriate floppy image or way of putting it on the floppy disk.
Have a good one.
--Dane