Usually the cylinder for Schlage doorknobs can be removed by turning the key in the knob and pushing a spring-loaded retainer to pull the knob off. On really old Schlage knobs this retainer wasn’t there, you had to “backdoor” the knob: completely disassemble it from the rear. This is much harder than taking the cylinder out of modern commercial knobs. In fact backdooring a Schlage A series was part of the test for the NWLA certified professional locksmith certification.
I can’t find any identifying information about this knob, I can only say that it is the predecessor to the modern a series. I put this here in case anybody runs into one and wants to know how to take it apart, the owner of this one told me that two locksmiths had told him it couldn’t be rekeyed. It absolutely can. The cylinder can even be replaced. I hope that these pictures help somebody!
To remove this vintage knob you have to slide that piece of metal out of the way 
To continue, bend the little tab straight and slide the metal housing off 
Here’s what the knob looks like underneath the cover 
Turn this piece of metal to the side to pull it off 
Push this bit to the side at an angle to get this hooked tab to come out 
Here it is ready to pull off the knob 
Pull this housing off carefully and don’t lose any parts 
Remove that object in the hole. Try pulling it out with something thin like a paperclip or lockpick 
Here is the knob without the round thing 
Now we can look into the knob and see the lock cylinder at the bottom 
Twist the knob while pulling the base up to remove the next part 
The next step is to move that object to the left away from the center of the knob. It is slotted into the long part that we will remove next 
Finally we have revealed the lock cylinder to rekey it 
Reassemble this knob the same way you took it apart. That’s why we take pictures, to use as a reference
