Bob;

Thanks again for taking the time to look this over. You are a treasure, sir.

There are many pellets which would roll about the same vertex of the cone.... You could have a .177 cal, a .22 cal, and a .25 cal which would all roll about the same vertex point, but the radius would be different, and a function of the caliber....

Yes, understood.

One cone which rolls about a different vertex (not the same vertex) could roll on a chord from the start point to the end point. But because we actually tie it to a point on a line which it must touch at two places that is prevented? It has to start on the same line, with the same angle. See attached. I think that is the point you make below?

In a similar manner, you could have vastly different pellets have the same starting point and ending point because they would follow a different route (radius) to get from where they started to a fixed end point, along different radii....

However, if they start sitting with one edge parallel (eg. against a ruler), they can only end up at the same point if "similar" tapers....

Also not sure I understand this. Similar tapers are defined along the entire length of the cone defined in the first proof in the thread. Any section of a cone which has a similar taper is by definition part of the same extended cone. They can not roll to the same place because the vertex upon which they roll moves with the starting point and they exist at a different radius from the vertex. The cone is also required to touch the starting line at r1 and r2 as defined in the first post. So one pellet from one portion of the cone can not roll on the same radius as one from a different portion of the cone when you establish the starting point and require them to be indexed to that point. The circle which a section of a cone describes when rolled is defined by the distance from the vertex whence it was extracted. Even if it has the same slope it's rolling radius is different. Proof 1.

That is why it is critical for this test to work properly to use a fixed point on the pellet (the back of the skirt is ideal) indexed to the same point each time.... If in your "exception" example the skirts were both at the "end" of the cone, they would still follow the same path.... They should do that if the difference in diameters, in relation to the length (ie the taper), and the skirt diameter, was the same....

I can see that it is critical for the pellets to be indexed upon some feature, the waist, or the base and further for them to be indexed on a line at the starting point, which should be parallel to the slope of the cone for the test to work. As far as I can tell there is only one way for a perfect machine to fail to distinguish between two cones. Perfect is clearly not achievable. Good enough certainly seems to be.

The example is the special case which only exists if the two "pellets" actually overlap in the cone which is probably pretty common in actuality. When they overlap in the cone the slope/taper (the tangent of the angle at the base of the cone) is the same for both sections.

In the previous examples we were discussing a a circle described by a line about the vertex, we are now talking about an area bounded by two concentric circles defined by r1 and r2 in proof 1. The distance between those concentric circles (h1 in proof 1) is the width of the gate they must pass. If one is fully contained within the other they can both pass through the gate only if the gate will pass the larger. If they only partially overlap only one can pass through the gate. If the gate is arranged so that one will pass the other will not pass. That presumes a perfect gate.

There is only one exception I can see. The special case I pointed out in proof 2.

Again thanks for your advice. I greatly appreciate it.

I am also attaching a picture @Yrrah gave me of his machine. He stacked those by rolling them.

Mike