This is a multi-volume set published from 1966 - 1973. Unlike many multi-volume family histories, there are different publishers for each volume. One should also note that the Family History Library filmed the typewritten manuscript of "Palmer families in America" in 1958, before the first volume was published.
The Family History Library has the typewritten manuscript, and vol 1 and vol 3 as published. Ancestry.com has volume 1 (1966 edition).
It is unclear whether or not v. 2 was published; I haven't yet found a reference to this volume.
Because each volume has a distinct publisher and editor of Mr. Horace Wilbur Palmer's research, there are three sources in WeRelate, one for each volume. As best I can tell, all three volumes are based on Mr. Horace Wilbur Palmer's research (in the typewritten manuscript of 1958) as compiled and edited for each volume.
v. 1. Lt. William Palmer of Yarmouth, Mass., and his descendants of Greenwich, Conn. / compiled and arranged by Horace Wilbur Palmer ; edited by Nellie Morse Palmer (Mrs. Horace Wilbur Palmer --
v. 3. William Palmer of Plymouth and Duxbury, Mass. / compiled and arranged by Horace Wilbur Palmer ; edited by Richard N. Palmer and Eunice (Wood) Palmer.