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Individuals H and J are 2nd cousins (supported by a strong documentation trail) who share two ancestors: John Charles Wright (circa 1830-1904) and Caroline Ellen Brown (circa 1827 – 1871). They share 223cm of DNA which is comfortably within the range of 2nd cousins (DNAPainter).

H and J have a common match (on Ancestry) D who descends from Henry Dillistone Brown born 15 May 1852 in Sawbridgeworth, Hertfordshire, illegitimate son of Caroline Ellen Augusta Brown (also supported by a strong documentation trail). They share 24.3cm which is (just) feasible for half-second cousins (which the documentation suggests.) (also DNAPainter)

I have double-checked the documentation trails and am happy that they are robust (including determining that the mother of Caroline Ellen Augusta Brown was resident in April 1861 at the address at which Caroline Ellen Brown gave birth to my ancestor in May 1861.) Although I'm comfortable that the two Carolines are the same person, would these DNA results unsupported by the smoking gun of the mother's residence be sufficient to conclude that Caroline Ellen Augusta Brown and Caroline Ellen Brown are the same person or would I need to look for further DNA evidence? If so, how would I go about it?

In my opinion, the DNA evidence - centimorgan amount, likelihood relationship, and triangulation - is suggestive but not sufficient. Two additional pieces of DNA evidence that I would look for include (1) DNA segment analysis or chromosome mapping and (2) clustering of shared matches through Caroline Brown and the parents of Caroline Brown. DNA segment analysis requires that DNA matches are on a site that has a chromosome browser and ensures that all matches share the same segments of DNA. Clustering would require more shared matches that are linked through Caroline or her parents as the MRCA. In addition, I think that most genealogists would want genealogical documentary evidence to support the relationship.

DNA rarely is enough evidence on its own.

The DNA results in this case support your theory that D is the half second cousin of both H and J due to an illegitimate child of a common great grandmother.

The DNA also supports other potential theories. For example, the half-sibling of H's and J's grandparents could come from the great grandfather. Or that 24cM match could come from any number of different relations. Half second cousin once removed on up and, on the full cousin side, even an 8th cousin could match with that amount (there isn't data for going further).

Your paper trail is strong enough to narrow this down to the one possibility you posit. But, if you didn't have that, your DNA evidence would not be able to narrow it down.

In this case you'd need a wide shot approach. Test everyone you can. Encourage every documented cousin you find to submit to Gedmatch (or allow you to do so for them) so you can run as many comparisons as you want, regardless of where people have tested. Eventually this will narrow down the family lines where this match can come from and you'll start to get some stronger possibilities.

MtDNA and Y-DNA may come to play a part as well. If your theory is correct then mtDNA will be the one that counts here, but Y could help in some other cases.

This is not an easy case to settle just with DNA evidence. But the DNA results can help you with the paper trail results.