You didn't say where this image comes from, but comparing it to another image I found on Google+, I am guessing that this refers to the material which has been published in electronic form in the collection of Rhode Island, Vital Extracts, 1636-1899 on Ancestry.com.
The Google+ news item shared by Ancestry.com said:
Published over a span of two decades, the Rhode Island Vital Extracts
(1636-1899) includes 21 volumes of details extracted from vital
records for the state of Rhode Island by James N. Arnold. Details were
taken from birth, marriage, death, baptism, church, and military
Their Source information says:
Ancestry.com. Rhode Island, Vital Extracts, 1636-1899 [database
on-line]. Provo, UT, USA: Ancestry.com Operations, Inc., 2014.
Original data: Arnold, James Newell. Rhode Island Vital Extracts,
1636–1850. 21 volumes. Providence, R.I.: Narragansett Historical
Publishing Company, 1891–1912. Digitized images from New England
Historic Genealogical Society, Boston, Massachusetts.
Beneath the Source Information box, there is an about the database collection describing the database and the nature of the information you might be able to find inside.
Note however that the original records came from NEHGS (who finished uploading volumes 7-12 to their online database in 2014). Here's an excerpt from the library catalog entry on their website:
Edition v. 1- Description 21 v. ; 30 cm. Contents v. 1. Kent
county.- v. 2-3. Providence county.- v. 4. Newport county.- v. 5.
Washington county.- v. 6. Bristol county.-
Going to the FamilySearch Research Wiki's article on Rhode Island Genealogy, we can get the history of the county formations in Rhode Island. Their table of Rhode Island County Creation Dates and Parent Counties shows the following:
- Providence County, renamed from Providence Plantations in 1703
- Kent County, established in 1759 from part of Providence County
This disagrees somewhat with the information from Wikipedia's article List of counties in Rhode Island. But about the town of Smithfield, Wikipedia says: "The area was within the boundaries of Providence until 1731 when Smithfield was incorporated as a separate town."
So my guess is that the first number refers to the number of the Volume in this 21-volume set, and the second number is the page number in that volume. I can't browse the images on Ancestry at the moment, but you can cross-check this hypothesis by using the browse function on the right-hand side of the page for the database there.
One thing you might try is to find a copy of Helen S Ullmann's book, A finding aid for Rhode Island town records in Arnold's vital records of Rhode Island, Beaman's Rhode Island vital records, new series, and the Rhode Island genealogical register. This link goes to the entry in WorldCat, which will show you if the book is in a library near you, by typing in your zip code.
I was not able to find copies of Arnold's Volumes 1 and 3 online, but looking at this Bristol County volume on Hathi Trust, I suspect that those volumes may have the same information as the page you show. If the parents and children belonging to the same family are indeed on consecutive pages (V 1, pages 50-51), then the section referenced here may be a compiled genealogy.
The information for the people whose references start with 3 may have come from a completely different record set; that might explain the jumps in numbers.
The difficulty with using any extracts of this kind is tracking down the original records that might have been the original sources of the information. In my own research, there is a large 2-volume history of a single family which hooks into my husband's family tree. There are thousands of names in the books, which took the author and his father over 25 years to compile and publish, and the only clue about where the information came from was that it came from "town records".
With all of these compiled works, the #1 question is "What did the author see?" It's a real challenge when the authors don't say, or the clues they give us to their source material are vague.
Update: This appears to be the same page at HathiTrust from this catalog item:
- Providence County [records of births, marriages and deaths].
- Main Author: Arnold, James N. 1844-1927,
- Language(s): English
- Providence, R.I., Narragansett historical publishing company, 1892.
- Subjects: Registers of births, etc. > Rhode Island > Providence
- Physical Description: 2 v. front. (map) 30 cm.
Perhaps the explanation for how the entire series is laid out was published in Volume 1?
According to genealogist Liz Loveland (via a Twitter conversation on 10-11 Mar 2015), it isn't unusual to see the same Volume and Page number for the same family:
Many New England town vital records recorded one family on one page if
they had room which would explain repeats (March 10, 2015)
Yes, it makes it quite handy when one is searching! Some clerks did it
that way & others wrote everyone's births chronologically. (March