This time we can't easily work out the fringe directions by marking out the points where the semi-circular ripples cross over. I've drawn ripples from four slits on my diagram but I could have added more. If I had done, the ripples from the top slit and the bottom slit would not even have overlapped. So what you have to do here is figure out directions in which there is constructive interference. We do this by drawing lines across the ripples in the way that Huygen's invented. This creates a wavefront where there will be constructive interference (you are joining up peaks and peaks!). To get the direction, draw a ray at 90 degrees to the wavefront. Notice that to reach the wavefront, the red wave has gone one wave further than the blue wave etc. Between the waves from one slit and the next on this wavefront, there is a path difference of one whole wavelength. 
The next diagram shows the same situation with the ripples removed. Since the rays travel (almost) parallel, the marked section shows the path difference. If the path difference is an integer multiple of the wavelength, you will get constructive interference in that direction.
Finally we extract the bottom triangle, marked X on the previous diagram and label it up as shown to get the diffraction grating equation.
