In a hour, hour needle run slowest. We counted how long after the cycle, minutes and seconds meet again.

We have:

For an hour, hour needle run 1/12 ring

For an hour, minute neddle run 1 ring

Each hour, minute needle faster at hour needle is:

1 - 1/12 = 11/12 [ring]

To meet again, minute needle, hour needle to run more than 1 ring. So the time to meet they again is:

1 : 11/12 = 12/11 [hour]

Same, we have:

For an hour, hour needle run 1/12 ring

For an hour, second needle run 60 ring

Each hour, second needle faster at hour needle is:

60 - 1/12 = 719/12 [ring]

To meet again, second needle, hour needle to run more than 1 ring. So the time to meet they again is:

1 : 719/12 = 12/719 [hour]

So, after: k*12/11 hour, hour needle and minute needle meet. 

m*12/719 hour, hour needle and second needle meet.

So, to three needles on a stright light after:

k*12/11 = m*12/719

=> k*719 = m*11

=> After law: 719*12/719 = 12 hour , three needles on a stright