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Hi

If he was 45 in 1879, that puts his birth at approx 1834.  However, I notice that on the 1880 US census for Pennsylvania, he was shown as 41 so born 1839. 

There are only two Joseph Riles on the 1871 census.  One was only aged 3 and the other was 17 b Islington London.  His occupation was woodchopper.  His parents were Thomas & Elizabeth and he had a sister Emma aged 13

I'm not very familiar with US death certs but thought they had the names of the parents on them.  Is this correct?

I found his daughter Margaret's birth on freebmd.  If you send for a copy of her birth cert, it will show her mothers maiden name.  This will enable you to trace the marriage of Joseph and Ann.  A copy of their marriage cert will give you their respective fathers names and their ages at the time of marriage

Margaret Riles quarter March 1878 barrow in Furness Lancs Vol 8E Page 863
You can order online at www.gro.gov.uk

Hi

From the info you give about George's siblings, I think the 1881 entry is the correct one.  However, George is shown as aged 5 which puts his birth at 1875/76 and his parents marriage 1873 onwards.
The 1900 US census shows a Harriet Burke b 1877 England working as a servant in New Jersey.  Immigration date 1894.  The census entry is faint but her year of birth looks like 1876 to me.  I can't find a UK 1891 entry for her


Dwelling:   28 East Robert St
   Census Place:   Salford, Lancashire, England
   Source:   FHL Film 1341946     PRO Ref RG11    Piece 3963    Folio 37    Page 23
   Marr   Age   Sex   Birthplace
William BURKE   M   25    M   Manchester, Lancashire, England
   Rel:   Head
   Occ:   Laborer Iron Works
Mary J. BURKE   M   25    F   Manchester, Lancashire, England
   Rel:   Wife
George W. BURKE       5    M   Manchester, Lancashire, England
   Rel:   Son
   Occ:   Scholar
Harriett A. BURKE       3    F   Salford, Lancashire, England
   Rel:   Daur
Dorothy BURKE       2    F   Salford, Lancashire, England
   Rel:   Daur
Dorothy BURKE       16    F   Salford, Lancashire, England
   Rel:   Niece
   Occ:   Out Of Employ


I assume you have already seen this on the IGI.

George William Burke 
  Male     
   

~~~~~~~~
Event(s):
 Birth:  05 JAN 1878   Manchester, Lancashire, England
 
 Christening: 
 Death:  26 JUN 1901   
 
 Burial: 
   

~~~~~~~~
Parents:
  Father:  William Burke   
  Mother:  Margaret Harriet Mary Jane Stone 

Hi

Here are the 1901 details.  RG13/3908  Folio 58 page 29
Address is 222 London ? road (bad writing)  Blackburn

As with the 1891, all are shown as surname Whalley.  On the 1891, Ruth is shown as b St Neots Cornwall and Selina in Blackburn.  On the 1901, the birthplaces have been reversed and Selina is shown as b St Neots.

The family is as 1891 except for an additional son - William A aged 9 b Blackburn

Hi

When you said their eldest child was b 1883, I assumed it was a first marriage.  As you said the marriage was possibly Liverpool/Southport, I assumed he was b Liverpool.  I found a William on the 1881 census born 1861 (so he would have been 22 when the first child was born)- see below.  This is the one I checked the image for.

You have now mentioned an earlier marriage and that he was b Stockport - facts not shown previously.   I can't find an 1881 entry for a William b Stockport but found the one below with a wife Elizabeth

Dwelling:   101 Upper Stanhope St
   Census Place:   Toxteth Park, Lancashire, England
   Source:   FHL Film 1341870     PRO Ref RG11    Piece 3635    Folio 39    Page 12
   Marr   Age   Sex   Birthplace
Jane CHALLENER   W   57    F   Llangefni, Anglesey, Wales
   Rel:   Head
Jane CHALLENER   U   26    F   Liverpool, Lancashire, England
   Rel:   Daur
Joseph CHALLENER   U   23    M   Liverpool, Lancashire, England
   Rel:   Son
   Occ:   Accountant
William CHALLENER   U   20    M   Liverpool, Lancashire, England
   Rel:   Son
   Occ:   Commercial Clerk
George A. CAIN   M   29    M   Liverpool, Lancashire, England
   Rel:   Son In Law
   Occ:   Printer Compositor
Martha CAIN   M   28    F   Liverpool, Lancashire, England
   Rel:   Daur
George A. (Jr) CAIN   U   > 2   M   Liverpool, Lancashire, England
   Rel:   Grandson



Dwelling:   11 Hart Street
   Census Place:   Westhoughton, Lancashire, England
   Source:   FHL Film 1341913     PRO Ref RG11    Piece 3821    Folio 60    Page 19
   Marr   Age   Sex   Birthplace
William CHALLENOR   M   26    M   Stapleford, Cheshire, England
   Rel:   Head
   Occ:   Coal Miner
Elizabeth CHALLENOR   M   22    F   Wigan, Lancashire, England
   Rel:   Wife
Samuel CHALLENOR       3    M   Aspull, Lancashire, England
   Rel:   Son
John Arthur CHALLENOR       1    M   Westhoughton, Lancashire, England
   Rel:   Son
Wright MOSS   M   52    M   Oldham, Lancashire, England
   Rel:   Lodger
   Occ:   Dateler In Coal Mine
William SHARPLES   U   21    M   Bolton, Lancashire, England
   Rel:   Lodger
   Occ:   Coal Miner