I have decided to settle this once and for all. Below I have set out much of the information that I know about this. I contacted the head librarian of the Royal Astronomical Society in London.
This is a copy of the following message you sent to Sian Prosser via Royal Astronomical Society
This is an enquiry email via https://www.ras.org.uk/ from:
[insert real name]
Dear Sian,
I am a member of the Zetetic Council for the Flat Earth Society (Yes, we are still going strong).
I am trying to find information about a paper on 'Zetetic Astronomy' that was read before the Council and Members of the Royal Astronomical Society, Somerset House, Strand, London (Sir John F. W. Herschel, President), Friday, Dec. 8, 1848.
No account of such a paper appears in the Notice for that month.
The paper was submitted by Dr Samuel Birley Rowbotham (pseudonym Parallax - author of Zetetic Astronomy: Earth Not A Globe - link as follows http://www.sacred-texts.com/earth/za/index.htm)
That book is one of the most important texts ever written for our society and the society Rowbotham built is the one we maintain today.
We know it was submitted for peer review but because it did not appear in the monthly Notice, we have no idea what the Royal Astronomical Society said about it. We have our own Flat Earth history archives and would love to add this information and share it with our members.
I do know that on Dec. 8, 1848, the Secretary of the Astronomical Society (De Morgan by name) said, at the close of the proceedings,—"Now, gentlemen, if you will promise not to tell the Council, I will read something for your amusement": and he then read a few of the arguments which had been transmitted by the lecturer.
Whilst I am sure my enquiry will be met with as much mirth today, as his appearance was then, the opinions of one of our greatest adversaries would be a real asset to our collection.
I understand you endeavour to help with enquiries 'as far as limited resources allow' but hope this slightly unusual request might pique someone's interest enough to check the archives. Alternatively I understand 'Access to the Library itself may be given at the discretion of the Librarian, or of one of the Officers of the Society, to those who have a need for information which cannot be fulfilled by public and national libraries.' I suspect I would have little luck in finding what I seek, but would be happy to try if such an invite was forthcoming.
Many thanks for taking the time to read this e-mail and regrettably, I have to add that I'm not crazy. I am one of many who find the history of our society fascinating.
[insert real name]
Member of the Zetetic Council for the Flat Earth Society.
And now we wait. I may have to take a trip to London to dig about for it myself. Anyhoo ... being as I have been very forthcoming with the information I have provided, I trust you will respectfully hold back your opinions until the documents are delivered or I have confirmation that RAS will not help me.
Interestingly, they do have an entire section called the Herschel Archives, and I hope that the document we seek is there. Their online library is very light and returns no search results for a number of key words.