I would suspect that the child born in 1838 has died, as it was very common to name the next child born, if they were the same sex, after the deceased child. It doesn't surprise me that you can't find a corresponding death registration because pre-1855 death registrations are fairly rare in my experience.
My 3x great-grandparents had two Christina's born in 1850 and 1852, two William's born in 1859 and 1861, and two Marion's, born in 1855 and 1874, the first Marion having died in 1871. The last two sets were relatively easy to work out because they were all post-1855 births and deaths. There was no death registration for the first Christina, but I could infer she had to have died sometime between her baptism in October 1850 and March 1851 because there was no record of her in the 1851 census. I finally got confirmation of her death from 1855 Marion's birth registration, because in 1855 the number of living and deceased children that the parents had had was included in the registration information, and this told me that they had had a deceased female child, which could only be 1850 Christina.
