This was not help for your problem. You must authorize both devices with the same Adobe ID. And have you eBooks buy before you should used the Adobe ID which has used for the first open. The Adobe ID is the Key to read the eBook. When you try to open a eBook with a other Adobe ID its not want work.
Thank you for your reply but I do not understand why Adobe will not let you delete your own ID.
However, I did what I mentioned in my original post and installed ADE 3 on my laptop. A box appeared asking me for the vendor (of the ebook I presumed) and the vendors login and password. After several unsuccessful attempts at using my library details, I used exactly the same Adobe details to authorise the laptop that I had used for the desktop. Instant success, so I will just use the laptop from now on as I cannot be bothered to spend any more time trying to find out why the desktop would not work. The ebook has now been successfully transferred to my Kobo Touch in EPUB form and I am enjoying reading it.
Its no problem from Adobe that you do need the same Adobe ID. This Adobe DRM are a wish of the authors and creator of the eBooks for the protection for the eBooks.