
I was experimenting on how to mount surface mount LEDs in wood and made this little arch to test out ideas.
All the LEDs are in series. But the question was how to drive them? I could connect them to a wall wart transformer and put in a resistor, but that would mean that the resistance of the resistor would have to be tuned to the wall wart transformer. Since I was just going to grab a transformer from my junk pile of dead electronics I didn't want to do that. Instead I settled on this constant current circuit shown in discover circuits:
http://www.discovercircuits.com/DJ-Circuits/constantcurretled1.htm
This circuit is really strange because the current from the LED goes through R1 along with the current from the LM334's current selection. Was it a typo? Should the LED current really just go straight to ground?
Also, I popped up the LM334 spec. The LM334 basically has 3 pins: Positive, ground, and a third "current selection" pin. You just connect a resistor across the current selection pin and the ground pin to choose what current you want. So it turned out that the LM334 is a constant current driver itself, so why have the extra transistor?
And In fact, the naked LM334 circuit what I ended up using. But the issue kept bothering me and so I finally decided to think about it.
I decided to try 3 simple circuits. Let's call the simple one "A":

So basically the current from the LEDs runs through the LM334, and so, regardless of the input voltage, is regulated by the size of the resistor connected between pins 2 and 3.
I guess the problem is that all the current is passing through the LM334...
And then a "simplified" discovercircuits, call it "B":

The LM334 is used as suggested by the spec. But the LM334 only controls the current that flows out of the base of the transistor, so current through the transistor and LED will be about 100 times (or whatever the amplification of your transistor is) what is going through the LM334.
This turns out to be very useful if you want to use the relatively low current LM334 chip to control a much larger current. Or if you only have larger value transistors lying around.
This one from discovercircuits "C":

As I previously said, "C" is really strange because the current from the LED goes through R1 along with the current from the LM334's current selection.
The LM334 holds is middle pin at 68mV above the bottom. So without doing any math, you can see that current going through the LED wil have to "share space" through the resistor. As the current through the LED rises, the current sensed by the LM334 goes down. This therefore is equivalent (to the LM334) like putting a bigger resistor across its pins. End result: the LM334 lets a lower current pass through itself (i.e. out of the transistor's base). You actually end up with a stable situation.
As with the second circuit (B), the advantage is that load is going through the transistor, not the LM334. An additional advantage is that the resistor values to make it all work are similar to that in the LM334 spec, and you don't have a 100x multiplication of the current (which may or may not be an advantage)
Another disadvantage of B is that it is very sensitive to the exact amplification (beta) of the transistor. But I guess transistor betas used to vary pretty widely within the same part due to small fabrication differences (I am not sure how accurate they are nowadays)... So while B is great for home use where you can test the exact beta of the transistor and choose a resistor to match, this circuit might be a bad idea to use in a production circuit board.
However, a big disadvantage of C is that the load (from the LED) is going through the resistor, so you do waste more power than in B and most importantly will have to use a high-watt (bulky, expensive, don't have it in your basement) resistor. Or of course a bunch of normal resistors in parallel...but that is also pretty ugly.