.why does the a stream of water get narrower as it falls from a faucet?
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The farther the water falls, the faster it falls. So 1 cubic cm of water is spread out over a greater vertical distance and hence the cross section of that volume must go down.
yes,The density of the water in contrast to the increasing speed at gravities constant acts upon the water in a inverse ratio? "i.e." the speed of the water vs. its cross sectional area density?
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The density of water is indeed constant. The volume is also constant. But the vertical height of a 1 cubic cm of water (or any other measure of a volume of water) will increase proportionally to velocity of the water. That means the cross sectional area must be inversely proportional to velocity of the water.
Velocity increases as the water falls farther. (Velocity will be proportional to the square root of the distance fallen.)
thanks for your help I got the numbers now from the above