Unless Mary had another daughter between 1860 and 1862, Clara must have married a HASKIN.  Or was Lottie actually James DOLAN's granddaughter, not Mary's?

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

Silas RASEY Birth: 24 Apr 1826, Ogdensberg, St Lwrnc, Ny; Death: 23 Jul 1862 Father: Joseph RASEY; Mother: Phoebe GREEN Marriage: 22 Apr 1854 Spouse: Mary E. BARHAM

2.  1850 Census Every-Name-Index/Images (online at Ancestry.com, Image #4-5 of 67):  Town of Gouverneur, St. Lawrence Co., NY, Roll M432_589, pp. 174A-174B, 19/19, enumerated 30 Jul 1850, official enumeration date 1 Jun 1850 (extracted by Diana Gale Matthiesen):

Silas Racey

24

M

Farmer

NY

Plus two other boarders, living with SHERWIN family.

3.  1860 Census Every-Name-Index/Images (online at Ancestry.com, Image #12 of 81):  Richville P.O., De Kalb, St. Lawrence Co., NY, Roll M653_853, p. 766, PN 12, 92/89, enumerated 15 Jun 1860, official enumeration date 1 Jun 1860 (extracted by Diana Gale Matthiesen):

Silas Racy	34	M		Laborer			New York
Mary E Racy	23	F		Housekeeping			Do
Charles E Racy	4	M					Do
Clara J Racy	2	F					Do
George F Racy	1	M					Do

4a.  1870 Census Every-Name-Index/Images (online at Ancestry.com, Image #58 of 85):  Adams Centre P.O., Town of Adams, Jefferson Co., NY, Roll M593_944, p. 29B, PN 58, 540/556, enumerated 13 Jul 1870, official enumeration date 1 Jun 1870 (extracted by Diana Gale Matthiesen):

Cummings Henry	56	M	W	Teaming			NY
________ Nancy	51	F	W	Keeps house		500	"
________ George	27	M	W	house Carpenter	800		"
Racey Mary E	33	F	W	Dress Maker			"

4b.  1870 Census Every-Name-Index/Images (online at Ancestry.com):  Son Charles is away at school (Cornell University in Ithaca).  There is a Clara RASSEY (æ 13, b. NY) living in Cohocton, Steuben Co., NY (p. 9A), with a millionaire and his family.  I can't find George.

5a.  1880 Census Every-Name-Index/Images (online at Ancestry.com, Image #1 of 23):  Adams, Jefferson Co., NY, Roll T9_838, p. 1B, PN 2, SD 7, ED 97, enumerated ca. 2 Jun 1880, official enumeration date 1 Jun 1880 (extracted by Diana Gale Matthiesen):

1880:  for an explanation of the column headings, please see What the Numbers in the Federal Census Mean (missing columns contained no data).
1	2	3	4	5	6	8	9	10	11	13	14	15	24	25	26
14	15	Comins Henry H	W	M	66			/		Farmer	3		NY	CT	CT
		______ Nancy	W	F	61	Wife		/		Keeping house			NY	VT	VT
		______ Geo Henry	W	M	37	Son	/			Carpenter	4		NY	NY	NY
		Rasey Mary E	W	F	43	Niece			/	Dress Maker	6	General ???	NY	Eng	VT

If Mary (BARHAM) RASEY is their niece and her mother was born in VT, Mary's mother is most likely Nancy's older sister (born before Nancy's parents moved from VT to NY).

5b.  1880 Census Every-Name-Index/Images (online at Ancestry.com):  can't find any of the children.

6.  1890 Census:  the 1890 Census Population Schedules were destroyed.

7.  1900 Census Every-Name-Index/Images (online at Ancestry.com):  Mary E. DOLAN (æ 63, b. Mar 1837, NY/Eng/VT) is living in Watertown City, Jefferson Co., NY (p. 41A), with her husband of 18 years, James DOLAN (æ 59, b. Jul 1840, Ire/Ire/Ire), and granddaughter, Lottie HASKIN (æ 18, b. Jun 1881, NY/NY/NY).

8.  1910 Census Every-Name-Index/Images (online at Ancestry.com, Image #2 of 9):  588 Albany St., Little Falls City (Ward 3), Herkimer Co., NY, Roll T624_952, p. 178B, SN 1B, SD 11, ED 24, enumerated 15/16 Apr 1910, official enumeration date 15 Apr 1910 (extracted by Diana Gale Matthiesen):¤•

1910:  for an explanation of the column headings, please see What the Numbers in the Federal Census Mean (missing columns contained no data).
		1	2	3	4	5	6	7	8	10	11	12	13	14	17	18	19	20	21	22	23	24	25	26	28
*	588	15	15	Rasey Charles F	Head	M	W	50	Wd			NY	NY	NY	Eng	Clergyman	Church	W	N	0	Y	Y		R	H
				Doron Mary E	Mother	F	W	70	Wd	4	4	NY	Eng	VT	Eng	None					Y	Y	Y
				Haskin Charlotte	Neice	F	W	25	S			NY	NY	NY	Eng	None					Y	Y	Y
*Albany St

9.  1920 Census Every-Name-Index/Images (online at Ancestry.com, Image #17 of 32 — indexed "Charles E A Rasey"):  G?? St., Little Falls City (Ward 2), Herkimer Co., NY, Roll T625_1115, p. 182A, SN 9A, SD 2, ED 29, enumerated 12 Jan 1920, official enumeration date 1 Jan 1920 (extracted by Diana Gale Matthiesen):¤•

1920:  for an explanation of the column headings, please see What the Numbers in the Federal Census Mean (missing columns contained no data).

1

2

3

4

5

6

7

9

10

11

12

17

18

19

21

23

25

26

28

*

???

???

???

Rasey Charles E S

Head

R

M

W

64

S

Y

Y

NY

NY

NY

Y

Clergyman

W

Dolan Mary E

Mother

F

W

82

Wd

Y

Y

NY

NY

NY

Y

None

Haskin Lottie E

Neice

F

W

36

S

Y

Y

NY

NY

NY

Y

None

*G???? St

This page image was faded and blurry.

10.  1930 Census Every-Name-Index/Images (online at Ancestry.com, Image #16 of 27):  West Ja??? Street, Village of Richfield Springs, Otsego Co., NY, Roll 1637, p. B, SN 7B, ED 39-48, SD 16, enumerated 23 Apr 1930, official enumeration date 1 Apr 1930 (extracted by Diana Gale Matthiesen):¤•

1930:  for an explanation of the column headings, please see What the Numbers in the Federal Census Mean (missing columns contained no data).

1

3

4

5

6

7

8

9

10

11

12

13

14

16

17

18

19

20

24

25

26

27

28

29

30

*

231

284

Rasey Charles E

Head

O

5000

R

N

M

W

74

S

N

Y

NY

NY

NY

Y

Local ti???

___ church

W

N

F2A?

N

Dolan Mary E

Mother

F

W

93

Wd

N

Y

NY

NY

NY

Y

None

Haskin Lottie C

Niece

F

W

48

S

N

Y

NY

NY

NY

Y

None

*

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

12.  Message posted to the Barham Family Genealogy Forum (online at GenForum.com).

13.  Message posted to the BARHAM Surname Board (online at RootsWeb.com/Ancestry.com).