Melbourne, Derbyshire comes under the Shardlow district.
There are 2 births for a Jane James in Shardlow, between 1856 and 1861:
Mar qtr 1859
Dec qtr 1860
There is also a death:
Dec qtr 1859
The 2nd Jane James (1860) is the daughter of William & Jane James of New Yard Melbourne (RG09/2489/60/13).
Do you have the marriage certificate of Frederick & Jane?
What I mean to say is: do you have incontrovertible evidence that they were, in fact, a couple?
The problem is I can't find Frederick with Jane in either 1861 or 1871.
But I can find Frederick with Caroline?!