This invention relates to the field of floating equipment for well bore casing.  When casing is being run into the well bore, particularly where deep wells are concerned, it is desirable to "float" the casing down to its intended location on the well bore fluid so as to relieve some of the strain from the derrick, prior to the time the casing is cemented in the well.  It is also desirable to have the casing fill automatically at a predetermined rate, so as to save rig time. 
To accomplish these desired results, "differential fill" float shoes and float collars have been developed, which devices permit automatic filling of the casing and also incorporate a back pressure valve to prevent cement back flow into the casing after the cementing operation.  Such a back pressure valve also permits the option of terminating the filling of the casing at any point in time.  An example of the prior art is the Halliburton Services Differential Float Collar and Differential Float Shoe, described and illustrated on page 3852 of Halliburton Services Sales and Service Catalog Number 41.  The collar and shoe employ the same valve assembly, which comprises a back pressure flapper valve at the top of the assembly, and a double flapper valve at the bottom of the assembly, the larger valve being a circulating flapper valve with a "piggyback" fillup flapper valve on it.  During the insertion of casing into the well bore, the back pressure flapper valve is held open by a pin set across the valve assembly bore.  As the casing enters the well bore, the preset spring tension of the fillup flapper valve spring allows controlled filling of the casing to a predetermined differential pressure between the casing interior and the well bore annulus.  Fluid may be circulated through the casing at any time due to the presence of the circulating flapper valve.  When it is desired to actuate the back pressure valve to prevent further filling of the casing as it is being run in, or after circulation has been established prior to initiating of the cementing operation for the casing, a weighted tripping ball is dropped, which breaks the pin holding open the back pressure flapper.  After cementing has been completed, the released back pressure flapper prevents cement flow back into the casing from the well bore annulus.  The above described valve assembly suffers from several noteworthy disadvantages.  First, there is a tendency of the pin to release the back pressure flapper prematurely, before the tripping ball is dropped.  Additionally, the ball's relatively unimpeded travel through the valve assembly resulted in the ball failing to strike the pin squarely or even missing it entirely, and passing downward without releasing the back pressure flapper. 
Another prior art differential fill float shoe is disclosed in U.S. Pat.  No.  3,481,397 to Baker, assigned to Halliburton Company, and incorporated herein by reference.  This design also possesses a back pressure flapper valve at the top of the shoe assembly with a double "piggyback" valve as described previously at the bottom.  The back pressure flapper valve is maintained in an open position by a short tube, slidably disposed in the shoe bore between the back pressure valve and the double valve.  The tube is maintained in place by a shear pin, and possesses a deformable lip at the bottom.  When the operator wishes to release the back pressure valve, a tripping ball is pumped down the casing bore to the lip in the tube, at which point the fluid pressure above the ball first shears the tube shear pin and moves the tube downward, releasing the back pressure valve.  After the tube has reached the full extent of its downward travel, the lip suddenly deforms, releasing the ball to continue downward and out of the shoe assembly.  The tube is maintained in its lower position by springs which snap over the top of the tube as it passes in a downward direction.  While an improvement over the above-referenced design, this device does not provide a full-open bore for the passage of cement in a downward direction, as the double valve is to be held open strictly by the downward flow of cement.  This poses problems due to jamming of the double valve by debris in the casing bore, and also reduces the immediate sealing pressure applied to the back pressure valve when the downward flow of cement under pump pressure ceases.  Furthermore, the means for maintaining the tube in its lower position are deficient in that they are exposed to the highly abrasive cement flow, with the attendant possibility of failure and jamming of the back pressure valve in an open position when cement pumping ceases and a reverse flow of cement commences.  Finally, the ball is subjected to such a great fluid pressure buildup above it when it contacts the lip in the tube, that the sudden deformation and downward release of the ball causes it to be "shot" at the double valve and break off the fillup flapper from the circulating flapper, or the latter from the shoe assembly. 
Other differential fill floating equipment designs are known which employ a sliding piston which holds the hydrostatic pressure in the casing to a percentage of the total depth of the casing run.  Such a design allows only one pressure, which on occasion causes excess floating of larger diameter casing.