Thanks for the information, Computer_Commando.
BC_Programmer, the computer I'm connecting the drive to to is a 200mhz computer, and is either a Pentium or Pentium II. (I'm not sure which one it would be, but it might be a Pentium.) The drive I'm planning on installing is a Samsung SD-612 DVD-ROM Drive 12X DVD-Master 12E OEM. The drive is being added, and will be the lower of the two disc drives, and will have to be the slave drive. (It has to be, as there's no other place to put the CD-ROM drive, which is a Goldstar 8x GCD-R580B. The reason why the CD-ROM drive can't be moved is because the front of its tray fits into the front bezel of the computer, which has its own CD-ROM open/close button.) I'm still planning on exchanging the ribbon cable, to be safe.
Below are pictures of the computer I'm talking about, an old HP Pavilion 7285 200mhz PC with the model number D3903A. (The PC is on the far right, near the counter.) This computer I don't use every day, and only use it if I'm going to play an old computer game from the mid-1990s. The bottom picture shows part of the bezel removed to allow a look at where the DVD-ROM drive will be installed. This is the same computer that a few years back, I had the CD-ROM drive at the top replaced due to problems with the original drive's tray opening on its own.
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