Hi, bm_brian16; welcome to our community!
Have you registered your subsctiption to your Norton Account, here:
https://www.mynortonaccount.com/
If not, go there on one of the computers that's already got one of your validated Norton 360 licenses running on it, and log in (if you have a Norton Account set up already) or click where it says "Sign up now" to create one. Once you've signed in, register your subscription (this may be done automatically, or you might need to use your Product Key).
When you have your subscription registered to your Norton Account, adding the new laptop will be easy: just log back in to your Norton Account on the new laptop when you have it, go to the entry for your subscription under "Products," and click the big yellow "Download" button. Once installed this way, your third license of Norton will automatically be activated under your subscription. It will then renew every time you renew that subscription.
Does that give you what you need?
V/R,
--DistEd2