I have a set of unpublished illustrations for a children's book (The Bee Man of Orn) dating from so long ago that I have no record or memory of the publisher or any contractual details. What can I do about getting them published? It seems a pity to have them languishing in a drawer.
When I originally read your question, I thought that YOU had illustrated the book and had some unused material left over. Having Googled the title, I find that this isn't the case.
My suggestion, contact the present publishers of the original book with a sequel or a parallel story which fits P.J. Lynch's unpublished paintings. If they're interested, they'll probably be willing to research and solve the legal aspects. Of course, your story line will have to be up to snuff.
You haven't provided enough details to really answer this question, espeically given that the book itself is old enough to be in the public domain (since 1976 in the UK). Broadly speaking, if they have definitely never been published, and the artist cannot be identified, they're copyright until at least 31 December 2039. They would be a current orphan work, and could be licensed as such.
You need to do some research, Larry, and find that publisher and/or the contract. They may still hold the rights to those illustrations. I'm assuming that as they were all done for one book, even though only a small number of them may have been used, they are all in the same style. Therefore they may be tied up in the contract.
Google it with your name and see what happens.