In
their early years, they used the logo seen on the left with
the plate number on the bottom of the logo (7836). We have posters
in our MPDB with this logo from 1928 until 1934. This one is
from the 1930 film Dawn Patrol.
In 1934, Continental changed from their logo
and moved to primarily just putting their name and plate number
across the bottom border, usually toward the middle or on the
left.
Here's the sample of the Continental tag
from the 1939 film Dark Victory. Notice the plate number at
the end tag line.

By the late 1940s, National
Screen Service was at full strength and had an enormous
need for the steady supply of plates for the massive amount
of posters produced. In 1957, NSS acquired Continental Litho
to supply the plates for them.
Once NSS acquired Continental, the name 'Continental'
was rarely used on the NSS posters. Instead you will normally
find just the plate number.
Unlike Morgan
Litho, Continental used a very unusual and complex numbering
system. We are reconstructing it now and will be putting a Continental
Number Log online soon.
For a list of posters currently in our MPDB
for Continental
Litho Co.
Since Continental was the lithographer for
Warner Brothers from 1928 on, here's our MPDB list on Warner
Brothers