Yep, you jumped straight to the top of the heap with that one! Ahrens Fox was considered the Rolls Royce of Fire Engines. The company started in the days of horse-drawn steamers, mid-1800s-ish. Large boiler used coal-fire to heat water into steam, which, given the tremendous expansion abilities of steam, gave great pressure. This pressure was, in turn, put to work driving the fire (water) pump. Usually, but not always, a piston-style pump.
Ahrens Fox developed a piston water pump which, through intricately casted pump housings, used one-way valves to allow the piston to pump on both the down AND up stroke, gaining more work, and more consistent pressure, in the process.
Now, as for that famous dome:
Picture a piston pumping water in a cylinder. As long as the piston is moving, it's pushing water out of the pump. BUT, every time the piston finishes its stroke, it has to STOP, and change direction. This creates quite a pulsing in the fire stream - not desirable for the nozzle man hanging on to a 2 1/2" hose pushing 250 gallons per minute through a smooth-bore nozzle at 50 psi. Enter the dome.
Water will not compress. Air, however, WILL. The dome is placed in the discharge manifold of the pump. When the piston is pushing water, some water is pushed into the dome, compressing the air in the dome. NOW, while the piston STOPS and changes direction, that compressed air forces the water in the dome back out the discharge. When the piston starts pushing water again, (stroking the other way), the pressure again compresses the air in the dome. This happens over and over, very quickly, as the pump is working.
The net effect is that the water discharge is much smoother, providing a steadier stream, and making it a little easier to hold onto the hose. It still pulses some. It's pretty cool to feel actually. (Today's centrifugal pumps use the spinning impeller to throw the water out, in a consistent, circular motion, ergo, no pulsing.)
Ahrens Fox used the same basic pump when they moved to motorized fire apparatus in the 1900s, and placed the pump in front of the engine (Yeah, I guess technically, it's a front-mount pump). The Ahrens Foxes were the most expensive fire engines, and had the best reputation, always capable of outperforming their rating. But all that hardware up front - the pumps were 3 to 6 cylinders - made the trucks not only incredibly long up front and very heavy, but harder than most to turn. Ahrens Fox sold out to the CD Beck company (I think) in the '50s, which in turn was bought my Mack. The first Mack cab-forward fire truck - the "C" model, was basically an Ahrens Fox design. In fact, I think two were made with the Ahrens Fox badge. These were, of course, centrifugal pumps by this time.
By-the-way, American LaFrance made steamers as well, both piston and rotary pumps. Their rotary pumps also used pressure domes (smaller ones) to smooth the water delivery.
When I get my first Ahrens Fox, I'll let you guys all drive it!!!

-Michael