While I was in and out of throwing together a response, Jim H beat me to the punch in I'm sure a more succinct fashion.
I seem to have a penchant for explaining the history of the calendar when you ask me the time of day

. But since I wrote it, here it is ...
I'm a better speaker than a writer
Regarding the PAP - This is an acronym for Positive Axis Point. An axis point is like the end of an axle. Since we are dealing with a ball, a more appropriate analogy is that the axis is like the one the Earth spins about, one point is the north pole and the other is the south pole. For arguments sake call the PAP the North Pole. Lets say the foul line is north (Positive) and the pins are south (Negative) or any way you wish.
Before I go on - To find yours (PAP that is), roll the ball with a normal strike release over board twenty (to catch a lot of oil). When the ball comes back, set it down so that the first ring of oil is parallel with the ceiling. The center of that circle is the PAP. These rings occur on the PINKY side - so remember
the P in PAP can also stand for Pinky

Once you've found your PAP, slap a piece of white waterproof tape there (you can also carve a tiny x there - foreign items on the ball surface are not allowed in competition, but a drilled an internally painted PAP marker should be okay).
Len Mal once told me that a bowlers PAP was like his signature.
A ball driller can explain it better. PAP coordinates are important assuming a consistent game
Observing the PAP is very important to understanding ball reaction or lack there of. Some examples, once a PAP totally migrates to zero degrees, and has lost any tilt (how high it points) the ball has rolled out or is rolling dead end over end (actually there is a tactic where roll out is good).
First of all, the axis being referred to is the initial axis the ball rotates about or spins around (just like the earth spins on an axis - so each axis point is like one of the earths pole, or the axle of a car or wagon). Other terms, some brought into bowling vogue by exotic core technology can muddy the waters for us simple folk (this ain't yer Daddy's pancake weighted ball) . Some very clear explanations are rendered in Don Johnson's second video on the release. Don's tapes were all pre-reactive era. His son (then 17) demonstrated what Don referred to as the urethane release. He also pointed out that one of his contemporary bowler's was nicknamed "Frisbee" because he employed some of the elements of the release that was popularized a decade or two later

it was either Karl Babb or Paul Colwell (apologies for spelling)
Both of Don's first two tapes are very good. The first place I ever read about it was in the Tom Kouros classic "Par Bowling" which I consider a bowling bible. Tom also described the difference between a flared track and a track that spirals - two different things. Flare is a tool to be used and managed. Bill Taylor's book balance exhibits the reason for why a track flares (then referred to as wobble as it was miniscule compared to today's meanderings) Very simply, think of the fact that the ball has a weight block the weight block is pulled on by gravity, as the ball spins or rolls, the ball will seek a stable axis, I think this is called preferred spin axis. Here's a baseball analogy, when a pitcher puts a dab of Vaseline in the ball, he's disturbed the distribution of weight (scuffing the surface is a different
gambit) Hey we should ask John Burkett - not that he doctors the ball but he does know both baseball and bowling.
Precession is not a term indigenous to bowling. If you were to see the Earth and it's moon, they actually travel a wobbled road in their combined path around the sun, in a manner of speaking, the moon is like a weight block because is forces that wobble.
Over on
http://ourworld.compuserve.com/homepages/kennmelvin/ you can witness via video the act of precession (the migration of said axis, i.e. the balls cycle of skid, change direction, roll - or however you wish to call it) The video in particular has a Storm Soccer ball, the Black Dot is that bowlers PAP. This is a good choice for one very good reason, the ball is not engineered to flare!!! Why is that a good reason? Simply because flare means multi tracking, since the tracks are NOT CONCENTRIC, the initial PAP will appear to wobble, sometimes this happens before the arrow area or before the bowler even sees it.
When a ball flares, the tracks have a common overlap area, this is referred to as a bowtie because it looks like a bow tie. Successive rings approach the finger holes. This is one very cogent reason why a ball driller needs to know how a bowler tracks, so as to prevent the tracks from traversing the finger holes, they have a (you guess it) term for a special drill safe zone (if you don't know the bowlers coordinates)
To complicate things, if the same bowler rolled three different balls with three significantly different weight blocks, they'd exhibit different tracks.
In fact, some balls delivered by strong enough bowlers flare in the air!
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Here's an old post of mine - there's some overlap but perhaps I was more awake then I am now - I'm a tad tired and may not be writing this very well ... time for beauty rest ...
Here's the older one -
I'm sure some one can do a better and more concise explanation, but I'll take the plunge and start this off.
First of all, the axis in question here is the one that the ball is rotating about. For a simple conception, of this just think about holding a water bottle in the palm of your hand palm facing up towards the ceiling so that the cap side is on the pinky side, the cap side is the positive axis point PAP, the bottom side is the negative axis point.
I think this issue gets a little muddy because of track flare and weight blocks, I'll skip that for now.
If you have a low flaring ball, lets use that for and example, if not, we'll use what you have.
Be that as it may, what we are looking for is the first track your ball makes when you roll a normal shot, if you are at the bowling center, roll the ball where you expect the most oil to be for the purpose of seeing the oil ring.
The PAP is equidistant from that oil ring.
By dead reckoning, you can estimate it and slap a piece of white tape on it (not in competition)
or if you have a measuring tape or other device
we can do some measurements.
I think this will get you close -
Set the ball on a table top so that the first ring of oil is parallel to the table top. Imagine that the oil ring is like the equator of the world (it won't be because it does not divide the ball in half for 3/4 rollers and spinners), what we are looking for is analogous then to the north pole.
So we measure from one side, going up over the top of the ball, use a grease pencil to trace that line, mark the halfway point, turn the ball 90 degrees, oil ring still parallel to the table top, repeat the measurement, where the two lines intersect should be the PAP, mark it with tape or a grease pencil.
Now with either method, when you roll the ball, if you have the PAP it will resemble the hub of a wheel and be fairly stable. As the ball starts to read the lane and flare, the PAP will wobble because now you are on another track that is not parallel to the first track and so on.
Sometimes if you really rev it up and the heads are dry, you'll never see that stable first PAP and would need an observer.
I've been told the PAP is like a bowler's signature.
The PAP is important because it is used in drill specs to accomplish reaction; you also need to know it to define the bowlers pin safe zone so that the ball won't flare over the finger holes; it defines the roll type, e.g. track type 3/4 roller spinner for example.
The orientation of the PAP defines the release, that is the vertical tilt of the axis, if it's higher, the track is lower. The lower the track the more skid, the higher the more roll.
A spinner has a low and small track.
The orientation of the PAP is used to control precession (the balls hook cycle sort of)
The PAP is used in discussing AOR axis of rotation. To simplify, if you rolled a ball 10 to 10 (foul line to arrows) and the PAP was perpendicular to the foul line that would be a 90 degree PAP, max skid, max flip. Your fingers were on the extreme side of the ball at release ) degrees would be a straight ball, your fingers were dead behind the ball at release - see this with the water bottle analogy)
You can change your AOR to suit the lanes.
Lets say the lanes are drier, you want a higher AOR, wetter, a lower AOR; there are other adjustments, this is just and example.
You'll see bowlers describe the PAP as so and so many inches over (from grip center midline (dividing top from bottom) I think) and so and so many inches or fractions of an inch expressed in vertical distant from that same line.
If you go to
http://ourworld.compuserve.com/homepages/kennmelvin/ and watch the video of the soccer ball, the black dot is the PAP there, there is almost no tilt if any and negligible flare (ball characteristics) but it shows the PAP and the act of precession, note how the PAP migrates, when it migrates it is reading friction, with no friction it would stay where it started. Look up precession on the web for a better explanation,
the earth precesses.
Precession is the act of the axis of rotation migrating to reach the axis or direction the ball is actually traveling (translational axis I think)
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If I had the time, I'd make this a bit more organized, but if you have the patience to have gotten this far, I hope it was worth the trip.
Take care.
P.S. Don't allow yourself to get overwhelmed by jargon. BTM has a glossary of terms. Many of the terms are terms of Physics which can also be researched elsewhere in non bowling websites for example. Each edition has ball reviews which detail the ball drilling with certain marks and terms. It's good to understand this because as you learn about yourself as a bowler and the conditions you bowl on
ball characteristics become very important. Yes surface chemistry and topography (scuffed, shined, whatever) are the dominant attributes, but weight block and position of block make up about a third or so of what the ball wants to do.
Years ago Rick Benoit wrote an article "what is your ball trying to do" Good article - I have it home in a file - used to be on a now defunct website.
For your perusal - periodically visit -
www.kegel.net for the tip of the month -
www.bowl4fun.com - Ron Clifton's tips -
www.beyondbowling.com Susie Minshew's website.
Hope this helps. Either way, I'd appreciate feedback so I can clean up my act if I need to ...
"Great minds discuss ideas; Average minds discuss events; Small minds discuss people."
Eleanor Roosevelt
Be well, John K. in Glendale Az.