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Hi AG
  Since i am weak in electronics i am getting confused . In the ckt the 3.9 Ohms is connected from o/p to adjust terminal so when the LED's are drawing 350mA current, then is the current flowing through the 3.9 Ohms resistor ? Or from where is it drawing current ?

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Heatsinks are not selected for their looks. They are selected by calculating how much cooling you need, then selecting one with the correct spec's. The manufacturer will say how many degrees C the semiconductor's case temperature will rise per Watt of dissipation when it is bolted to the heatsink. It is assumed that no insulator is used and that thermal compound grease is used between the semiconductor and the heatsink.

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So I would have to use a 6 volts stepdown 500mA transformer,bridge and a filter which would give an o/p voltage of 8.64 volts dc across the filter capacitor to reduce the voltage to lm317 ?

Almost. The rectifier bridge reduces the voltage to 7.1V. Then the max output voltage of the LM317 will be about 4.6V and the 3.9 ohm resistor reduces the max voltage for the LED to only 3.35V.
Doesn't the LED need 3.9V?
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Hi AG
        I connected a 11 Ohms resistor as the load . I noticed that the o/p voltage of the lm317 regulator was 3.6volts dc when the input voltage to the regulator was varied from 7.82volts dc to  17volts dc . The lm317 is cooler then before now . Is this enough AG  ??? as the forward voltage for the led is 3.48 volts dc ? 

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Hi AG
        Assuming the current drawn by each LED is 350mA . I have connected a 500mA transformer for the whole circuit . Now to make 5 LED's work together, Will it be good to put them in series with a transformer of more o/p voltage? Because the i/p to the regulator is maximum 40 volts i guess !! I am not sure. Will it be safe ?

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I have this graph that shows the amplitude of peak-to-peak ripple voltage (at double the mains frequency) but it is for my 60Hz mains and is wrong for your 50Hz mains frequency.
The ripple reduces the average DC voltage by half the amount of the p-p that is on the graph.

For your 350mA then a 1000uF capacitor will have about a 2Vp-p ripple which is fine. Just add 1V to the minimum input voltage of the LM317 current regulator.

post-1706-14279143514166_thumb.png

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Hi AG
        Actually I made the circuit on bread board and by mistake i connected the capacitor(1000/25)in reverse but nothing happened to it . Was it supposed to blast ? But it was almost hot by the time i came to know my mistake and switched it off .

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