The 1861 Census details are:
1861 Census - Overgate, Lawsons Close, Dundee, Forfarshire
John Carmichael 42 Slater b. Kinross, Orwell
Helen Carmichael 33 b. Forfarshire, Dundee
James Carmichael Son 10 b. Lanarkshire, Glasgow
William *Dawson Boarder 3 b. Forfarshire, Dundee
Alexr. Carmichael Son 1 1/2 ditto
*Transcribed on Ancestry as Dawson but have viewed original on SP and doesn't look like Dawson to me, possibly Lawson - definitely ends in 'son'.
1851 Census - 202 (?) Overgate, Dundee, Forfarshire
Helen Carmichael 25 Slater wife b. Forfarshire, Dundee
James Carmichael 1 b. Lanarkshire, Glasgow
Elizabeth Duncan 19 Sister Unm. Mill Worker b. Forfarshire, Dundee.
This raises a number of questions:
1. Looks as if 'William' wasn't the natural son of John and Helen - clearly shown with a different surname (be it Dawson or something similar) and a 'Boarder' on 1861 census. I can't see on birth indexes that John and Helen had a son William of their own. They did have a son Andrew Duncan born 22/8/1857, died 1858 born around the same time as William. Perhaps they took in William having just lost their own child - whatever the reason it certainly would seem that he was not their own child although perhaps he was never told this and believed them to be his parents.
2. James Carmichael was bc.1850 in Glasgow and I'm guessing the age of 13months is an error on ships list for 1864 and should read 13 rather than 13 months. He was certainly alive in 1861 and there is no death entry for him between 1861 and 1864.
3. Cannot find John Carmichael on 1851 Census - just his wife Helen with son James and sister Elizabeth Duncan. If Helen was Couper/Cooper how was her sister a Duncan?
So, there is no trace of John and Helen having a son William ca.1858 which is when they did have a son Andrew Duncan b.1857, d.1858 but they did have a William Dawson (or similar) who was aged 3 and a Boarder in 1861. Seems too much of a coincidence for William Carmichael in 1864 not to be this person who was probably raised from an infant by John and Helen - sort of an unofficial adoption. This, of course, puts a spanner in the works as it seems that William wasn't actually a Carmichael at all and his actual surname is difficult to read on the census. If you buy some credits for Scotlands People you can view the 1861 Census image yourself and you can see for yourself the difficulty on reading his surname. Births index shows no William born in Dundee as Dawson and I don't believe the name is Dawson - possibly Lowson, definitely ends 'son'.
Sorry to mess things up!
Annette