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I think what your looking for is the 400 Jeffery's is the equivalent of the 450/400/3" without a rim. the 450/400/3" is a rimmed cartridge.

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Sort of.
They use different diameter bullets - .408-.410 for the 450/400 3", and .423" for the 404 Jeffery Rimless.
The 404 was developed as a comparable rimless version of the well proven 450/400 3" for use in bolt action rifles.
I don't know why the 404 was designed with a .423" bullet but I suspect that it was actually developed in Germany and swiftly adopted by Jeffery as their proprietary cartridge much the same as they did with the .500 Jeffery later on.

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The information below is from from research undertaken by Casey Lewis & Alf Smith who wrote 404 Jeffery - The Full Monty published in Big Bore Journal No. 21.
Although Jeffery consulted Krupp in regards barrel specifications, it seems the cartridge and bullet design was an all England initiative.


"Jeffery did not have access to magnum length Mauser actions and
had no choice but to use the standard Mauser action. In doing this
he could not use the 450/400-3" case turned into rimless as the degree
of opening up it needed would have affected action integrity too
much.
Jeffery had to design a shorter rimless cartridge by reducing the case
length of his popular .450/400x3" case from 3" 76.2mm to 73mm.
This still marginally reduced the safety margin, but was sufficiently
strong as long as the pressure was kept at the original specification of
17.2 tons per square inch (2,240 lbs to the ton). This considerable
reduction in the volume of the cartridge case would have resulted in
a drastic increase in pressure had they retained the .450/400x3" load
and kept the same diameter bullet as for the .450/400; namely the
.408".
Jeffery worked through Le Personne with Krupp regarding barrel
specifications as well as Eley and Kynoch who possessed the cartridge
design expertise, to duplicate combustion space by increasing the case
base diameter from 13.76mm/0.5417" to 13.843mm/0.545" and
the shoulder diameter of the .450/400x3" from 13.233mm/.521" to
13.462mm/.530". The rimIess version's bullet diameter was consequently
increased to .4225". The nett result of the modifications to
the .450/400x3" was the shorter, fatter, rimless cartridge we know as
the .404 Jeffery and which used the weight of bullet and propellant
as its rimmed predecessor."

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