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Hi Michael,
I am the originator of the this thread back in July 2017. I am also a member of the 'Guild of One Name Studies' ( www.one-name.org ). The surname I am studying is 'Audley'. I believe that the 'Adley' surname is a variant of the 'Audley' surname (or vice versa) particularly when looking back to County Galway in Ireland. There are of course other origins to both surnames.
All my research into the Audley Surname (including how it links to the Adley surname) is freely available on both of my websites. I suggest you look at www.audley.one-name.net and look for 'Audley Family M' download the 'Audley Family M' file and search for {M279} I believe he is your Grandfather, and I believe {M301} is your father.
Living in England I have little access to US records after the 1941 census and whilst I know a little of your family, it may help you recover the information that you lost. I would also appreciate any information and or photographs, that you would be prepared to allow me to place on my website. It is only with the help of people like you, and responders to my questions on Rootschat that my website has become what it is.
If you wish to discuss this further that is great, we can either do it through Rootschat or if you contact me through my website and then we can exchange e-mails.
Regards
Brian

Hi Sandra; SBrevoort & floppyslater

Many thanks for your replies.
I  have now definitely concluded that I have found the same family in both the first and second enumerations of New York in the 1870 census even though in the first enumeration the family is transcribed as AUDLEY and in the second the family is transcribed as ARDLEY
One of the statements in one of the links provided by Sandra is very informative as it states:
' Curiously, upon first examination the two enumerations of New York City bear little resemblance. The explanation lies in the fact that between enumerations the election districts changed, making it difficult to compare the two.
 This explains why the family I am interested in was in were in the 29th election district at the first enumeration and the 4th election district in the second enumeration. I think I understand it all; I just hope I remember next time I look at the 1870 census of New York.
Thanks to all for your help

Hi RJ137,

I have been pondering over your reply for a number of hours and I am now wondering whether both census returns are correct.

With regard to the 1870 census return for Catherine ARDLEY on the Ancestry website it states above the image  ‘ 2nd Enum (Second Enumeration) and the top of the image it gives an enumeration date of 18th January 1871. Whereas in the 1870 census return for Catherine AUDLEY the top of the page gives an enumeration date of blank day of July 1870.

This would mean that on the first enumeration ( July 1870) only Francis was living with his parents;  whereas in the second enumeration (January 1871) only Mary was living with her parents. It is possible that Mary could have been omitted from the  first enumeration (July 1870) as her brother fills the last line on the page and they omitted to include her at the beginning of the next page.

 In support of this theory the Family directly above the AUDLEY family in the first enumeration are the FAUCETT Family and in the second enumeration they are the family BELOW the ARDLEY family. The head of the FAUCETT Family is on the same page as the ARDLEY family with the rest of the family on the next page.

Equally on the First enumeration starting on line 6 is the Stacom Family Head Francis Stacom followed by his wife Ellen Stacom followed by 9 children. In the the 2nd enumeration Francis and Ellen Stacom appear on rows 26 and 27 on the following page after the ARDLEY family.(their children are not included in the second enumeration

Would anybody agree with my logic that the family (AUDLEY/ ARDLEY has been enumerated twice once in July 1870 ad then again in January 1871.

Treetotal,
Thanks for your message
Regards
The Wirral Way

Thanks to all for your messages.

A number of you have made reference to the 'Cracker Jack Dixie Cup Military Cap' including a link to Humphrey Bogart wearing such a cap.

To me looking at several images of such a cap, it appears to me that this Cracker Jack Cap has a wide brim completely around the the circumference and this brim points upwards (see the photo I have attached). This is very different from the hat being worn by Joseph Audley in my original message

Regards
The Wirral Way