Which is correct, Green or Green's function?

The predominant usage of Green function is in the possesive form (i.e., Green's function). As a physicist, I find myself conforming to this as well, however, I was corrected by my Ph.D. advisor who always insisted that that was incorrect. "It is Bessel function, not Bessel's function, and so it should be with Green." (I am paraphrasing her here.) Is this true? I notice that Jackson (of Classical Electrodynamics) refers consistently to Green function, as does Eyger and Cohen-Tanoudji. Others (Fetter and Walecka, Arfkin, Morse and Fechbach, etc.) consistently use Green's function. Many others, I have found, interchange them, as do you, e.g. Physics Today 2003 uses Green function throughout, while your review of Mary Cannell's biography of Green uses Green's function.

Where is the first citation where the term is coined? I've read Green's work and he doesn't coin the term himself. I presume it was Lord Kelvin, but I cannot find a reference for this. Is there a consistency in the symantics in that first publication?

While this is an inconsequential issue, it has been one that has irritated me for some time. I would be very happy to put this topic to rest once and for all in my eyes.

Also, what would be your recommendation of the best George Green biography for a physicist to read?

Thank you,

TJ Ulrich, Ph.D.

Los Alamos National Laboratory