I'm just guessing that SMART was Jane's surname based on the name of her daughter.

1.  Marriage Record:

2.  1800 Census Index/Images (online at Ancestry.com; Image #54 of 64; extracted by Diana Gale Matthiesen):

1800

NC

Rutherford Co.

p. 146

Ln. 13

Adam Thompson

00100-10010-00

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
1 male	16-25	1774-1784	= Adam (b. 1773)
1 female	9 or under	1790-1800	= Daughter A
1 female	26-44	1755-1774	= Jane (b. 1775-80)

Adam and Jane should be in each other's age class, yet surely these are our subjects.  Although the list was roughly alphabetized and there were eight THOMPSON (and two THOMASONs) in the county, Adam was listed next to Elizabeth THOMPSON, who was surely his widowed mother. 

3.  1810 Census Index/Images (online at Ancestry.com; Image #8 of 66; extracted by Diana Gale Matthiesen; see 1810 Table):¤

1810

KY

Christian Co.

p. 60/520

Ln. 1

Adam Thompson

30010-20010-01

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
3 males	9 or under	1800-1810	= William (b. 1805) = George (b. 1802) = John (b. 1801)
1 male	26-44	1765-1784	= Adam (b. 1773)
2 females	9 or under	1800-1810	= Martha (b. 1809) = Matilda (b. 1807)
1 female	26-44	1765-1784	= Jane
1 slave	all ages	in or bef. 1810	= ?

4.  1820 Census Index/Images (online at Ancestry.com; Image #18 of 19; extracted by Diana Gale Matthiesen; see 1820 Table):¤

1820

KY

Trigg Co.

N half of Co.

p. 13B

Ln. 21

Adam Thompson

020201-12020-0-300

no slaves

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
2 males	10-15	1804-1810	= William (b. 1805) = George (b. 1802)
2 males	19-25*	1794-1801	= John (b. 1801) = ?
? male	26-44	1775-1794	= ?
1 male	45 or over	in or bef. 1775	= Adam (b. 1773)
1 female	9 or under	1810-1820	= Jane (b. 1810)
2 females	10-15	1804-1810	= Martha (b. 1809) = Matilda (b. 1807)
2 females	26-44	1775-1794	= Jane = ?
*In the 1820 Census, the third column is age class 16-18 and the fourth column is age class 16-25; therefore, any individual in column three is duplicated in column four.  By subtracting the number in column three from the number in column four, you can create an age class "19-25."

There are no children in the 1810 Census who could be the unknown individuals in the 1820 Census.  They must be boarders or other kin, etc. 

5.  1830 Census Index/Images (online at Ancestry.com; Image #29-30 of 58; extracted by Diana Gale Matthiesen; see 1830 Table): 

1830

KY

Trigg Co.

p. 16

Ln. 22

Adam Thompson

000 010 010 - 000 110 010

010000-000000

These data indicate:

No., Race, Gender	Age Class	Therefore Born	Individuals Inferred
1 white male	20-29	1800-1810	= William? (b. 1805)
1 white male	50-59	1770-1780	= Adam (b. 1773)
1 white female	15-19	1810-1815	= Jane (b. 1810)
1 white female	20-29	1800-1810	= Matilda (b. 1807)
1 white female	50-59	1770-1780	= Jane
1 male slave	10-23	1806-1820	= ?

6.  1840 Census Index/Images (online at Ancestry.com; Image #30-31 of 67; extracted by Diana Gale Matthiesen; see 1840 Table): 

1840

KY

Trigg Co.

pp. 291B-292A

Ln. 10

Adam Thompson

000 100 001 - 000 000 100

no slaves

0100000

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
1 white male	15-19	1820-1825	=
1 white male	60-69	1770-1780	= Adam (b. 1773)
1 white female	40-49	1790-1800	=
1	persons employed in agriculture

7.  1850 Census Index/Images (online at Ancestry.com):  can't find.  Jane does not show up in Trigg Co., KY, as head-of-household, nor is she living with any of her three daughters.

8a.  A.B. Willhite.  [n.d.]  Christian Co., Kentucky, Tax List, 1804-1809.  (self-published; available from author at 4396 Coopertown Rd., Russellville, KY 42276, or from the Christian County Genealogical Society, Hopkinsville, KY):

Pg. in Willhite	Year	Dist.*	Last  Name	First Name	A	B	C	D	E	Acres	Watercourse
25	1804	North	Thompson	Adam	1	-	-	-	4	200	T W
48	1805	North	Thompson	Adam	The records were so damaged the transcriber simply listed the names and did not take any numerical data.
66	1806	North	Thompson	Adam	1	-	-	-	4	200 200 200	T W T W T W
110	1807	West	Thompson	Adam	1	-	1	1	4	200 200 200	T W T W T W
-	1808	-	-	-	-	-	-	-	-	-	-
200	1809	West	Thompson	Adam	1		1	1	4	200 200 200	T W T W T W
Legend for numerical data: A = White Males 21 or over; B = White Males 16-20 (this column not used in 1809); C = Blacks over 16; D = Total Blacks; E = Horses
Abbreviations:  T W = Tradewater River
*The county was divided into just two districts:  from 1804 to 1806, into North and South Districts; beginning in 1807, into East and West Districts.

8b.  A.B. Willhite.   [n.d.]  Christian Co., Kentucky, Tax List, 1810-1813.   (Self-published; available from author at 4396 Coopertown Rd., Russellville, KY 42276, or from the Christian County Genealogical Society, Hopkinsville, KY).

Pg. in Willhite	Year	First  Name	Last  Name	A	B	C	D	E	Acres	Watercourse
42	1810	Adam	Thompson	1		1	-	4	200 200 200 400	T W T W T W T W
-	1811	-	-	-	-	-	-	-	-	-
143	1812	Adam	Thompson	1		1	1	-	200 200 200	T W " "
-	1813	-	-	-		-	-	-	-	-
Legend for numerical data: A = White Males 21 or over; B = White Males 16-20 (this column used only in 1811); C = Blacks over 16; D = Total Blacks; E = Horses
Abbreviations:  T W = Tradewater River

8c.  A.B. Willhite.   [n.d.]  Christian Co., Kentucky, Tax List, 1814-1818.   (Self-published.  This book is out of print.  I do not have a copy.  If you have one to sell, please contact me.)

8d.  A.B. Willhite.   [n.d.]  Christian Co., Kentucky, Tax List, 1819-1823.   (Self-published; available from author at 4396 Coopertown Rd., Russellville, KY 42276, or from the Christian County Genealogical Society, Hopkinsville, KY).

Pg. in Willhite	Year	First  Name	Last  Name	A	B	C	D	Acres	Watercourse
65	1819	Adam	Thompson	1	-	-	5	100  70 100 200 200 200	M F " " - - -
-	1820-23	-	-	-	-	-	-	-	-
Legend for numerical data: A = White Males 21 or over; B = Blacks over 16; C = Total Blacks; D = Horses
Abbreviations:  M F = the Muddy Fork of the Little River

Adam drops off the Christian Co. tax rolls in 1820, when Trigg Co. was formed.  The Muddy Fork land (in Trigg Co.) was purchased in 1818, and while he still owned the Tradewater River land (presumably in Christian Co.), it appears he that genuinely moved in 1818/19 because he shows up in the 1820 Census of Trigg Co..  In other words, he wasn't in Trigg Co. "all along," as one might think based solely on the censuses.

9.  Christian Co., KY, [Deed?] Book A, p. 216 (courtesy of Ruth Call).  Document dated 26 Aug 1806, in which Adam Thompson of Christian Co., KY, son of Samuel Thompson of Rutherford Co., NC, deceased, gives Elizabeth Thompson, Executrix of Samuel Thompson's estate, power of attorney to sell the tract of land in Rutherford Co., NC, where Samuel had lived and died.

10.  Christian County Genealogical Society.  1997.  Christian County, Kentucky, Deed Abstracts of Books G-H-I (1816-1819).  Christian Co. Gen. Soc. (1101 Bethel St.), Hopkinsville, KY 42240.

p. 141	DBI-496, Oct. 30, 1818, from ROBERT GOODE to ADAM THOMPSON, for $500.00, 100 A. on Muddy Fork Creek, in metes and bounds, S/Robt. Goode, Elizabeth Goode, Wit: Wm. Porter, John Goode, Dower Relinq., Rec. 31 Oct, 1818.
p. 148	DBI-567, Dec. 7, 1818, from JOHN GOODE to ADAM THOMPSON, for $350.00, 70 A. two roda and 19 poles, on the Waters of Muddy Fork of Little River, bounded Cain's line, Williams' line, others. S/ John Good, Wit: Benjamin (his X mark) Ladd, John (his X mark) Ladd, Rec. Dec. 7, 1818.

11.  Lon Arneld Bostick, Sr.  1987.  County Court Order Book "B" Part 1 for Christian County, Kentucky, for the years 14 Oct 1805 - 12 Sep 1809 for a County Court held at Town of Hopkinsville.  Christian County Family and History Research Society (4465 Cadiz Rd., Hopkinsville, KY 42240).  First column is the page number in Bostick; second column is the page number in Court Order Book B.

p.	p.	Date	Motions
072	154	6 Apr 1807	On motion of James Bradley for a road leading from James Wilsons at or near James Scoggins horse mill ordered that James Wilson, William Stroud, Adam Thompson, Thomas Waters
	155		Samuel Bradley and James Bradley or any three of them being first sworn do view and mark the same and report thereof to our next county Court.
075	164	1 May 1807	A Bill of Sale from John Gray to Adam Thompson for a Negro woman named Jude produced in Court acknowledged and ordered to be recorded.
108- 109	234	Jun 1808	On motion of James Bradley ordered that Adam Thompson, James Bradley, James Reeves and James McKnight or any three being first sworn do view and mark a road from Temple West to James Reeves on a direction to the house and barn.
131	288	13 Feb 1809	Ordered that the viewers heretofore appointed to review and mark a road from the forks of the road near Reeves to the forks near Adam Thompsons have the further time until next County Court to view and mark the same with M. Early and Abm. Hodge additional viewers.
133	294	13 Mar 1809	The viewers appointed last Court to view a new part of a road from James Reeves to Adam Thompsons made their report to Court and on consideration it is ordered that the said road be established.
138	308	8 May 1809	Ordered that the road as heretofore viewed from the Livingston road near Johnson's field to the Hopkins County line on a direction to Mencers horse mill be established and that Captian (sic) George Crismill be appointed surveyor of the same as far as his house and that William Metcalf be surveyor from that to the crossing of the Land Lick Fork of Trade Water and Adam Thompson from that
	309		to the County line and that Joseph Clark, Esq., allott the hands and return an allottment to next County Court and that this road be cut 15 feet wide.
145	323	14 Aug 1809	On motion of Jacob Morris for leave to alter the road leading from Judge Wilsons to the Livingston County line as far as from the forks of the road between Adam Thompsons and said Morrises to the house and barn it is ordered that James Cook, Matthais Early, William Howard and Jesse Hail or any three being first sworn do view and mark the same and report to next County Court.
147	329	11 Sep 1809	The viewers heretofore appointed to review a part of the road leading from the forks of the road near Adam Thompsons to the house and barn made their report to Court and on consideration it is ordered that said review
	330		be set aside.

12.  Barney Thompson. MS. The Thompson Family of Prince Edward County, Virginia, Butler and Trigg Counties, Kentucky, and the Florida Parishes of Louisiana:  the Descendants of Thomas Thompson, 1749-1810.  Copies at Western Kentucky University, the Filson Club Library, and the Indiana State Library, Indianapolis.  (Excerpts courtesy of the author, with my sincere thanks for generously sharing them.)  The following excerpts are of articles written in the 1890s by Cyrus Thompson for a Trigg County newspaper called the Kentucky Telephone.  They are an octogenarian's reminiscences of life in early Trigg County.  (Adam Thompson is not related to Cyrus Thompson or to the Thompsons who are the subjects of Barney Thompson's book.)  In Chapter 8, under the heading "Cyrus Thompson Articles, Part Two" (boldface added):

p. 60

I remember with some distinctness many others of the old settlers who lived in early times in the vicinity of Cerulean Springs-- from 1832 to 1840-42.  The forms and features of some of them I can now see in my mind's eye as I saw them fifty years ago, and well remember their social and moral standing.  The Goodwins, the Goodes, the Pools, the Sizemores, the Stinebaughs, Rogers, Guthries, Adam Thompson, Elisha Harbor, David Haggard, John Jones, and the Wilsons, were all good men-- amongst the best citizens of the county.  The heads of most of these families came to the county in the early part of this century, and nearly all from Virginia and North Carolina.  There were all farmers, nearly or quite all were slave owners, and they were all tobacco growers.

p. 61

Adam Thompson I bear in mind distinctly.  He was a devout Christian-- a member of the Christian Church-- one amongst the earliest converts to that religion, and Sunday after Sunday would ride on horseback in bad winter weather from his home to Cadiz to "break bread" and partake of "the wine" with his brethren, of whom Wayman Crow was one, and at whose house he always stopped.  He, poor man, had an only son, William, a promising young man who died in Cadiz in about 1835, and it nearly broke his heart.  He had studied medicine and had just entered upon the practice with bright prospects of success.

13.   William Henry Perrin, ed.  1994.  History of Trigg County, Kentucky.  Heritage Books, Bowie, MD (this book is a facsimile of just the Trigg Co. portion of Perrin's Counties of Christian and Trigg, Kentucky: Historical and Biographical printed in 1884 by F.A. Battey, Chicago).  To read the entire section on Cerulean Springs Precinct, click here. 

p. 010	Settlement by White People.	After the Goodwins came ... Adam Thompson ... all of whom became citizens and had homes in the northern part of the county as early as 1800.
p. 121	Cerulean Springs Precinct.	Later came ... Adam Thompson ...

14.  LDS.  Family Search: Internet Genealogy Service:  AF - Ancestral File (online at FamilySearch.org).

15.  Emails from Ruth Call, Barney Thompson.