I have not been able to find F. S. Tull in any other Confederate organization, nor the officers of his original company.
The National Park Service National Battle Field Park at Pea Ridge have a marker with three guns of Tull's Battery. I
would suggest you contact the Park Historian to see what information they have on the battery.
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Pea Ridge Order Of Battle
Missouri State Guard
2nd Division, Brigadier General Martin E. Green
Various unidentified infantry and cavalry units Unknown
Kneisley's Missouri Battery (5 guns) Captain James W. Kneisley
3rd Division, Colonel John B. Clark, Jr.
1st Infantry Major John F. Rucker
2nd Infantry Colonel Congreve Jackson
3rd Infantry Major Robert R. Hutchinson
4th and 5th Infantry Colonel J. A. Poindexter
6th Infantry Lt. Colonel Quinton Peacher
Tull's Battery Captain Francis M. Tull
http://www.civilwarlandscapes.org/cwla/states/ar/pr/intro/oobf.htm
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Pea Ridge National Military Park - National Park Service - U.S. Department of the Interior.
3. Tull's Missouri Battery
The cannon here represent Captain Francis M. Tull's Confederate Battery. The battery occupied a line in this vicinity on the morning of March 8, 1862. Three 12-pdr Napoleon guns and three empty carriages fill out the display. Tull's battery was armed with smaller 6-pdr guns at the battle.
http://www.hmdb.org/marker.asp?marker=35672
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Here are three other call outs for the battery, but I have not been able to gain any further information trying to track.
Tull’s Battery “1st Rifle Battery”
Bentonville, Pea Ridge, Van Buren (1862) [Assume these were engagements]
http://www.arkansascivilwar150.com/civil-war-sites/regiments/display.aspx?side=Confederate&state=MO
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Tull’s Missouri Light Artillery, Capt. Francis M. Tull (Consolidated with Bledsoe’s Missouri Light Artillery)
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Tull's MO Light Artillery Battery
Organized:
Assignments March 1862 to May 1862 [nothing further]
Brigades Artillery, Frost's, Frost's/Clark's
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Interesting document from the Official Records, the only one mentioning him.
MONTGOMERY, April 8, 1861.
Governor J. J. PETTUS,
Jackson, Miss:
The artillery company of Jackson, of which R. H. Purdom is lieutenant; the artillery company of Vicksburg, of which F. S. Tull is captain, and the artillery company at Natchez, of which William S. Lovell is captain, are needed at once at Pensacola. Give them orders and let me know the result. There is work for them.
L. P. WALKER.
OR V52, Pt. 2, p. 39