Angular dividing is all about prime numbers. Somewhere in your worm gear and hole plate you need ALL of the prime numbers in your desired division. 39 is a composit number and contains the prime number 3 and 13 (3 X 13 = 39) so you must have the prime numbers 3 and 13 somewhere. None of the common worm ratios will have 13 in their prime number complement but some may have the 3. So you will have to have either a 13 hole circle with the factor of 3 in the worm or a 39 hole one with the more common 40::1 worm ratio. Hence you need a hole circle with either 13 holes or some multiple of that (13, 26, 39, etc.) or one with 39 holes or some multiple of that (78, 117, etc.)
You can either buy one or MAKE one. Buying one may be problematic if your dividing head is not a common brand as there is no guarantee that a hole circle from one brand will fit on another brand.
There are several approaches to making one. The first thing you need to have in mind is the angular accuracy that your job needs. There is no sense in wasting time in a commercial shop making one that is better than what you actually need. You need to make a spline so I think you may not need the ultimate accuracy.
I will estimate the resulting accuracy but you did not mention what worm ratio your head has so I will assume the most common one which is 40::1. If yours is different, then the accuracy estimates will be off. The common principle here is the fact that a worm drive in a dividing head or rotary table functions in an accuracy amplifier when you use a plate with a hole circle. This means that if you have a circle of holes and they are located +/- 0.5 degrees and you use that to make a part OR ANOTHER HOLE CIRCLE WITH THE SAME NUMBER OF HOLES, then the error will be reduced by a factor of 40 with a 40::1 worm. So your resulting error would be only 0.5° / 40 = 0.0125° or about 45 seconds of arc. And that is pretty good. Of course, you are also limited by the accuracy of the worm itself. Keeping this in mind, here are some methods for making your 39 hole circle, in order of increasing accuracy.
For all the methods you must first make a blank disk that will fit on your head.
Method 1: Using manual layout techniques, lay out a circle on your blank disk and then manually divide it into 39 divisions. This can be done with layout die and a drafting compass. Take a guess at the needed division and then step around the scribed circle 39 times. See how far off you are off and adjust the compass by about 1/39 of that amount and try again. Do this until your 39th step lands on your starting point. Then do it for real, scribing a small arc that intersects the scribed circle. Using a magnifier and a prick punch, locate each hole as accurately as possible at the intersection of those 39 arcs. Then carefully punch then and finally drill them with a new drill. A reamer for final size would be an additional touch. With care, you should be able to locate those 39 holes to an accuracy of around +/- 0.003". If you used a circle of about 3" diameter, the radius would be 1.5". Using trig, the angular error in your manually constructed hole circle would be about sin(0.003/1.5) = 0.115°. And if you use that hole circle, the worm's 40x reduction in that error would yield an error of about 0.0029° in your spline. That's about 10 arc seconds. My 10 inch RT is not that good and I doubt that the worm in your indexing head is either.
Method 2: This is much the same as Method 1 except you can use a CAD program to lay out the hole circle for your blank. With care, this will probably result in a similar level of accuracy as Method 1. The sources of error are a bit different, but some will be greater and others will be less so they will probably average out and you wind up with that same final error in your spline.
Method 3: This will yield a hole plate with greater accuracy than Methods 1 and 2. I doubt that this is necessary, but the resulting plate could be used for direct indexing instead of using the worm. Many indexing heads allow for this and it can be a time saver if a lot of parts are being made. Instead of making just one blank plate you make two or three. The first one can be a lot sloppier than with Methods 1 and 2. Use that first plate to make a second generation one which will have only 1/40th of the error of the first. Then, a third can be made with the second one and it will again be 40 times more accurate. If you start with a plate with an accuracy of only +/- 1° for the first plate, using rough, manual methods then the second generation plate will be +/- 0.025° and the third generation plate would be +/- 0.0006° or 2.25 arc seconds. I will guarantee you that the worm in your indexing head is no way near that accuracy. It is about as accurate as you can do with your head.
That's three ways to make a hole circle with an accuracy that you may need for your part.
