I'm including the link here to credit the author, Harry Mentakos, as well as pasting the contents in case the link dies. Literally, the last sentence of this write up summarizes what most of us do, ignoring everything the article describes. That's also the way I learned this in previous squadrons, and I suspect what most will do anyway when flying TACAN penetrations. However, this write up is for those of you who want a more precise technique break down. Additionally, there are a ton of videos online that can walk you through this as well, and spare the read for those of you who prefer a visual vs written description. As always, note that this is a method, not a directive.
How to Fly a DME Arc
If you center the OBS needle with a FROM indication, then the number at the top of the OBS dial tells you what radial you're on. If you wanted to fly a circle around the VOR, you'll need to fly perpendicular to that radial, on a course that's 90 degrees from the radial. Of course, as soon as you begin to fly a course that's perpendicular to a radial line, you'll immediately find yourself on a slightly different radial, and to trace out a perfect circle, you'd now need to change your course to be perpendicular to that radial. You could do all this math in your head, or you can cheat and notice that these headings (there are necessarily two of them, one for a clockwise circle and another for a counter-clockwise circle) are on the OBS dial at the 9 o'clock and 3 o'clock positions. e.g. if the OBS dial is tuned to 360 with a centered needle, then you'll see 090 and 270 on the left and right edges of the dial). If there was zero wind, and you wanted to fly a perfectly curved circle, you could do this by continually adjusting the OBS dial to keep the CDI needle centered, and continually flying the heading that's 90 degrees from that radial.
That sort of "continual heading correction" is a higher workload than necessary, though, as there's really no requirement to fly a perfect circle. The PTS actually allows your distance from the VOR to vary by up to 1 nm (e.g. on a 10 DME arc, you just have to keep it between 9 DME and 11 DME). So the conventional approach is to break up the circle into 10 degree chunks, and fly a series of straight lines. In other words, if you were going to do a full 360 degree circle (not that you would in the real world), you'd break that into 36 straight line segments. If you flew this perfectly, you'd be flying a 36-sided polygon (a "triacontakaihexagon", of course), which is pretty close to a circle, at least in terms of its ability to ensure that you remain on a path that doesn't intersect with a mountain.
So draw a circle. Take two points on that circle that are about 10 degrees apart, draw a line that connects them. If your points are on the circle itself, then the line necessarily ducks inside the circle a bit, but not very far. Note that the heading of this line isn't a perfect 90 degrees from the radial at either "intersection point", if it was it would have to be parallel to the circle at the points where it touches (and a line like that can only touch the circle in one place), but the only time this line is actually 90 degrees from the radial it crosses is at its mid-point between the two ends, while it's inside the circle.
So at the beginning of this line, you're actually turned 5 degrees "inside" the circle, and at the end of this line, you're actually turned 5 degrees "outside" the circle. So for the first half of the line, you're slowly getting closer to the VOR, and for the second half, you're slowly getting further away.
So, at the start of a line segment, in order to determine the course for that line segment, you figure out the radial you're on, then figure out the course 90 degrees from that (not using arithmetic, but by looking at the OBS dial), then you turn to a heading that's 5 degrees towards the inside of that heading (e.g. 5 degrees to the left if you're going counter-clockwise, or 5 degrees to the right if you're going clockwise).
Let's try flying this. Imagine you're 15 miles north of the VOR on the 360 radial, flying a heading of 180 to intercept the 10nm DME arc. You track that 360 radial in towards the VOR, keeping the needle centered, until the DME reads 11 or so (i.e. you're about a mile outside the arc). Twist the OBS dial 180 degrees until you have the needle centered with a FROM indication (stop tracking the CDI needle at this point, just hold your heading constant for now, as the needle will be reverse-sensing for this little step). Since we've been tracking our radial perfectly, we now have a centered needle and the OBS says 360 with a FROM indication. Now, at the 3 o'clock and 9 o'clock positions on the OBS dial are 90 and 270. Let's say we need to track the arc clockwise, so we know we have to make a left turn to start, so the correct heading to join the arc must be "roughly" 90 degrees. That's the first hurdle (a lot of people make that first turn in the wrong direction, and get headed counter-clockwise when they mean to go clockwise). If we turned to a heading of exactly 090, though, we'd be slowly heading outside the arc, we want to start our "line segment" heading 5 degrees inside the circle, so that we (ideally) next intersect the circle at the next radial, 10 degrees around from here. If we picture flying a circle clockwise around a VOR, then "inside the circle" must be to our right. So our first heading (again, reading off the OBS dial) is a left turn to 095.
Note that if we had wanted to fly counter-clockwise, we'd take the 9 o'clock number off the OBS, which is 270, "inside" the circle would now be towards our left, and the course that's 5 degrees to the left of 270 is 265, so we'd turn right to a heading of 265 to join the arc. If it helps, you can think of 9 o'clock as being counter-clockwise and 3 o'clock as being clockwise from the top of the OBS dial, and you choose the course from the corresponding side of the OBS dial for flying counter-clockwise vs. clockwise arcs.
Also note that that we chose our initial heading based on the OBS, and that it had nothing to do with our inbound heading, i.e. we didn't just take our inbound heading and subtract 90 from it. Our inbound heading isn't necessarily always perpendicular to the circle, for instance we could (in theory) have been getting vectors to intercept the arc, or we could (in theory) be tracking a radial off a different VOR, etc.
So you know your "next thing" is going to be a turn to 095. Wait until the DME reads 10.5 (a half mile outside the arc), and then begin your standard rate turn to 095. (BTW, don't wait until it's time to do the next thing to figure out what that next thing is, by then it's too late). Once this turn is complete, adjust the OBS dial for the "next" radial. We joined at 360, so going clockwise our next radial must be 010, so we dial the OBS to 010. Now we should see a right deflection of the CDI, and as we fly it will gradually get closer to center (this would go the opposite way, i.e. a left deflection slowly moving to center, if we had flown the arc counter-clockwise).
Once the needle centers, we know we'll be on the 010 radial, and it will be time to start the next line segment. Before that happens, we need to figure out what our next heading will be, so we're ready to do it as soon as the needle is centered. As we did when we joined the arc, we look at the 3 o'clock position on the OBS (or 9 o'clock if we're going counter-clockwise), and read the heading that's 5 more degrees towards the inside of the circle. Since our OBS is currently set to 010, the heading 100 is at 3 o'clock on the dial, and 105 is 5 degrees to the inside of the arc (inside is to the right on a clockwise arc, and to the left on a counter-clockwise arc), so our next heading will be 105. As soon as the CDI needle centers, we begin our turn from 095 to 105. We just passed the 010 radial, so our next radial must be the 020 radial, and we now advance our OBS dial to 020. We continue this process of "turn ten, twist ten" until we're ready to exit the arc.
Now, all of this stuff is really calculating courses to fly, and if we just take a course and use it as our heading, we're ignoring the wind. So how do we correct for the wind? Well, by trial and error, of course. As you're flying along the arc, you monitor your DME reading, and try to keep it as close to 10nm (in this example) as possible. A headwind or tailwind component will just speed you up or slow you down, but you're not timing anything, so we don't care about this. A cross-wind component, though, will tend to blow you either outside the arc or inside it. So if you see yourself drifting outside (the DME reading is getting higher than 10), you need to adjust your heading towards the inside of the circle. As you fly the arc, you need to keep present in your mind at all times which direction "inside" is (it will be to your right when going clockwise, and to your left when going counter-clockwise).
There are two ways to adjust for DME "drift". One is to do the above machinations, and apply an explicit wind correction angle to your calculated course. For instance, if the OBS tells you your next course is 015, and your DME is getting too high (the wind is blowing you outside the circle), you could try a heading of 020, which is 5 degrees to the inside of your "OBS course". If your DME reading is still trending the wrong way (i.e. it's too high, and getting even higher), then increase your wind correction angle, e.g. start adding 10 degrees to the heading the OBS gives you. Note that the correct wind correction angle will continually change, as your course changes as you fly around the arc, but as long as you seem to need that correction to adjust for the wind or to correct a deviation, continue to do the "turn 10, twist 10" dance, and keep applying your (guessed) correction to the new course that the OBS tells you (e.g. 5 degrees to the inside of the 3 o'clock, plus your wind correction).
If your DME reading is too low, i.e. the wind is blowing you inside the circle, then instead of applying an explicit wind correction angle, you could alternately just skip or delay making a heading change. Note that every turn you make when crossing a radial is a turn towards the inside of the circle, and if you delay making that turn, you'll be implicitly adjusting your course to the outside. You could continue to delay updating your heading until you're back on the desired DME reading, and then "rejoin" the usual procedure. Note that if there was a _very_ strong cross-wind, this might not be enough to keep you reasonably close to the arc. But for small deviations to the inside, this is a sensible first try. For instance, it's a reasonable policy to say "no point advancing to the next heading while my DME is still too low", but if delaying your heading change isn't enough for your DME to start getting closer to the right value, then put in a wind correction to the outside.
While the PTS says you need to keep the DME within 1nm of the arc, it's always easier to make small corrections earlier than large corrections later, and there's no reason why you shouldn't be able to keep within .1 or .2 of the correct DME for the arc.
(One wind-correction "cheat" for the GPS-equipped could be to adjust your heading until your ground track equals your desired course. In other words, make small adjustments to your heading until your ground track matches the course you determined from the OBS dial (plus the 5 degrees to the inside). Getting your ground track from a GPS is a tremendous help for tracking radials, localizers, etc., but keep in mind that there's typically a significant "lag" in the ground track reading, it doesn't give you instantaneous readouts of your ground track).
For all this to come out right, you need to keep the sequence straight:
pick your next course by looking at the OBS dial at 3 o'clock (flying the arc clockwise) or 9 o'clock (flying it counter-clockwise), then adjusting that heading by 5 degrees to the "inside" (right for clockwise arc, left for counter-clockwise arc). Note that you always pick your heading when the OBS is dialed to the radial that you're just about to cross (or, when entering, that you tracked in).
turn to that heading when the needle centers (or, when entering, the DME is .5nm before the arc)
advance the OBS by 10 degrees
lather, rinse, repeat
When we exit the arc, we'll most likely either track a VOR radial inbound or outbound from the VOR. In our case, let's say we're supposed to follow the arc until the 061 radial, which we then track towards the VOR. To intercept that radial without passing it, we need a "lead radial" to use as an indication of when to begin the turn inbound, so that by the time we've completed our turn (about 30 seconds at standard rate), we'll be on that radial, and not have blown past it. Let's pick a lead of 5 degrees, so we'll use the 056 radial as our lead radial (i.e. our "signpost" that tells us to start our turn here so we'll be on the right radial once we're done turning). While we're tracking the arc, when we pass the 050 radial, instead of advancing the OBS to 060, we'll just set it to 056 instead. Now, when the OBS needle centers, we'll know we're on the 056 radial, and can now begin our turn to a heading of 241, which is the reciprocal of 061 (in other words, to track inbound to the VOR along the 061 radial, you must be flying a course of 061+180 = 241). We then reset our OBS to 241, and track the needle inbound.
Note that there are alternate ways of doing this. The technique above has the OBS at full deflection after you update it as you cross a radial, then making your next turn when the needle centers. An alternative is to go from a half deflection right to a half deflection left (or vice-versa, for counter-clockwise). This scheme can make it easier to determine the heading to turn to, because it will be right on the 3 o'clock or 9 o'clock positions, you don't have to "add 5 to the inside". But to enter it "gracefully", your first step is a little special, as you move the OBS to a heading that's only 5 degrees ahead, instead of 10.
There's also a "no-brainer" approach that has you mostly ignoring the OBS, and just watching the DME, making 5 degree turns left or right to try to keep the DME at the right number (crude, but possibly effective).