Final answer:
The year the Noncommissioned Officers Manual was published is not provided in the available resources; further research in military records or historical documentation would be necessary to find the specific year.
Step-by-step explanation:
The Noncommissioned Officers Manual was published by the War Department and is part of the historical documentation of the U.S. military, but the specific year of publication is not provided in the resources available. Key historical documents that might be associated with the military or Noncommissioned Officers, such as Executive Order 9981 which ended segregation in the military, or the Northwest Ordinance are important, but they do not specify the publication year for the Noncommissioned Officers Manual. To get an accurate date, one would need to consult additional military records or historical documentation that directly references the manual.