battery question: matching batteries to motors

I think I have a path for you to figure out the numbers. If I quote wrong data (I am doing this from memory) I am sure someone will correct me.
First I assume you do not have access to a dynamometer. Yu need to figure out how much power it takes to do what you want to do. I believe a normal human can produce 100 watts continiounously and about 400 to 500 watts for a short term peak,
First go ride your existing bike at a steady comfortable speed. Do this for 6 min (.1 hour) and record the average speed. A recording speed app on a gps or smart phone would be useful. This gives you the speed that 100 watts generated.( you being the power source. let us assume this is 10 mph
Second ride your bike as fast as you can for 6 min and record the the speed. This gives you the speed you can get with 400 watts. let us assume this is 20 mph
You now have 3 points, 0 MPH/0 watts/hour, 10mph/100 watts/hour ad 20 MPH at 400 watts per hour. You can either hand plot these on a piece of graph paper or find a nice curve fit for excel. This will give you a graph of how much power it takes to move your bike at a given speed. Also the power you need in watts to go the speed you want (will the motor generate that much power)
Lets assume the speed you want is 10 mph ( since this makes the numbers easy, you will of course use the numbers for 40 mph.
You need 100 watts per hour. Assumn a 48 volt power pack, this means V x A = Watt or Watt\ V = amps so you need 2.1 amp hours to go 10 mph. 10 mph would take you 17 hours. 17 hours x 2.1 is about 36 amp hours. So in this case you need 4 of the packs you quoted.
Note this will only be a swag, but will give you a starting point to see if this is even doable. Any errors here are my fault from bad memory. This does not take in account legal questions, can you even run an electric assist bike at that speed, Depth of discharge and other issues which will be useful at final purchase. This will only give you a broad estimate of what you need.
Good Luck

1200W motor is huge and will need a large battery to feed it = heavy - for a bicycle at 40mph = not good!
1200W/48V = 25A current draw, but it may be even more than this - check what the controller maximum current output is!
Most batteries are rated at 1C discharge so as a minimum you will need a 48V 25Ahr battery. The battery should not be drained below 80% DOD for battery longevity, so that battery would give you about 45 minutes at 25A draw.
Most of the other data you need to calculate the sizing is available on the endless sphere forum but that site seems to be down at the moment.

I want to travel 170 miles and I'm looking at the Cyclone E Bike website and their 1200W Double Freewheel kit. How do I figure out what size battery to buy that will run that distance at at least 40mph.

Will a 48V 10Ah LiMn battery work? Could you explain why or why not this will work, please.

Doing 40 Mph requires at least 40 Wh/mile. So, for 170 miles this would require 6.8 kWh of energy. For the given 48V and taking into account some losses, this would mean at least 150 Ah. The answer to your question is: No the 10Ah will definetely not be sufficient. At that speed the battery would support you for about 12 miles.