The number 1 reason people fall on downhills: Letting a hand get behind you. If your hand goes back, your body twists. If your body twists, you loose pressure on the front of your skis. If you loose pressure, your skis go out from under you. Crash!
Keep the hands in front of you at all times. Keep them in sight. They can move up and down or in and out depending on how much balance you need - but keep them in front!
On turns, it's the outside hand that's important. Notice Hugh skiing around tightly set pylons in the sequence below. In each photo, his outside hand is always in front:
Let's take another look at a left turn:
In both sequences above, Hugh is always facing in the direction of the turn. His chest and hips square with the turn. By putting your outside hand in front of you, you guarantee and that your shoulders and hips will be in the right place. If you left you outside hand lag behind, so will your shoulders and hips.
A dryland demonstration
Try this:
Stand up facing forward, feet about shoulder width apart. (Yes, right now!)
Twist your shoulders and hips as one unit to the right. Notice what your feet do: they roll on edge to the right.
Now twist your shoulders and hips as a unit to the left. Your feet roll on edge to the left.
If you did this on skis going down a hill, your skis would automatically turn in the direction you're facing - they'd be on the correct edges.
What happens if your outside hand goes back? Pretend you're turning left. Twist to the left - your feet roll to the left. Now throw your right hand back behind you:
Your shoulder twists back to the right, pulling your hips with it
Your "edges" release
There's no longer any edge pressure forcing the ski to turn
You're probably leaning back slightly, taking weight off the front of your foot
If you're on skis, you're in a very precarious position indeed! There's a tree with your name on it just waiting for you!
Recovery
If you get in the situation where your hand is too far back, there is only one way to recover - if you have time: Punch your hand forward!
Turning in a Tuck
Want more speed? Get into a tuck. The rule of hands forward applies even more. Notice the sequence below. Hugh's hands are in front of his face, leading him into the turn. Being in front, they pull the shoulders and hips in line with the turn, allowing Hugh to keep pressure on his edges.
Hugh is stepping around the corner - in fact, he's probably taken three steps already through this short sequence. In the middle image, his outside foot is lagging slightly behind as he steps onto the inside ski, but his hips are still in the direction of travel.
Summary
The direction your shoulder and hips face determine what edges your skis are on.
Use your hands to control the direction your shoulders and hips face. Where your hands go, your shoulders and hips follow.
Hands in front keeps your weight in a neutral position over your skis. If your hands go back, so goes your center of gravity.