Answered By: Reed Lowrie Last Updated: Jul 15, 2015 Views: 55
I think you'll find what you're looking for at the link found at this HOLLIS link:
The interface is not the most friendly, but it looks like the volumes from the 1750s are available in full text.
It may not have turned up in a search because the 'official' title is just Philosophical Transactions. I searched 'transactions royal society' in the title field and 'london' in any field.