Ah .. the 4.0.10 version is only for OS 8 & 9. And should only be installed whilst actually booted into those operating systems (if possible.) I think it can then be run whilst OS 9 is booted up as classic environment in OSX however if desired, though might not run as well as being booted up into OS 9.
You will find 4.12.15 version link just near the other link:
Lime Wire » Download Out-Of-Date Installers. It depends which operating system you wish to run LW on.
You did not state which OSX version you are running.
If you wish to use 4.0.10, you should update your OS 9 to 9.2.2 & the equivalent java for best results.