This is an explanation on how to add new thread when the thread in your shuttle (or ball) has run out.
When I need to add new thread, I usually do it at the end of a chain, and the next element to be tatted is a ring. Here, the shuttle thread is looped round the fingers to make the next ring. The old thread end is laid along the shuttle thread while the start of the new thread is pinched together with the shuttle thread at the beginning for the ring.
Make the first half-knot for the ring but do not slide it right up to the beginning. Instead, after flipping it, keep it loose and pass the old thread end through the loop. Make sure that the end passes through the loop in the same direction as the shuttle thread. Then, hold the shuttle thread and the old thread end together and slide the half-knot to the beginning of the ring.
Make the second half-knot. Again, after the flip, keep it loose and pass the old thread end through it. Remember to pass it through in the same direction as the shuttle thread. Hold the shuttle thread and the old thread end together and slide the second half knot to completed the doublestitch.
Make a few more doublestitches in the same way, passing the old thread end through the loop at each half-knot. This is sometimes described at tatting over tails, i.e. the doublestitches are tatted over the thread ends.
Right, we still have one other end to deal with, i.e. the end of the new thread. The same treatment is done for the new thread end, i.e. tatting over it in the next element which can be either another ring, or in a chain.
In the piece that I am making here, my next element happens to be a chain. If the chain has only a few doublestitches, say only four or five, followed by a ring, I will continue tatting over the ends into the following ring.
I usually leave the ends in the piece and not cut the excess thread immediately, in case I make a mistake and need to undo it. That way, I still have enough length of the old thread to work with.
Another suggestion is to block the piece first and then cut off the excess thread. Normally, tatting thread will shrink a bit when the piece is moistened for blocking. This shrinkage will further tighten the doublestitches over the thread ends and secure it further.