Rebecca or Sarah Jay?

In tracing the Emmerson line down through the Suffolk area, Mendham to be precise, I uncovered the following conundrum:

1891 Census, George Emmerson (c1844, Mendham) listed with wife Sarah A R Emmerson (c1847, Mendham) and daughter Louisa L M Emmerson (c1887, Mendham).
Marriage register: Sarah Ann R Jay.
In 1901, the only George born 1844 in Mendham is married to Rebecca (c1846, Mendham), with daughter May (c1887, Mendham).
In 1881, Sarah A R Jay is listed with mother Lydia.
In 1861 and 1871, Rebecca is a servant and then visiting, respectively.
In 1851, Rebecca S Jay is listed with parents Lydia and Edward.

There
is no matching May Emmerson listed in 1891, and no Louisa Emmerson listed in 1901. However, a birth record in Hoxne (registration district for Mendham) lists May Lydia L Emmerson in Oct-Dec 1886.

My conclusion is that these girls have a penchant for being know by both christian and middle names and that these are one and the same family.