The theory can be stated as follows:
"If an object is thrown back and forth between two people, and the maximum height of the object in flight remains constant, the duration of the flight remains constant regardless of the distance between the people."

So, if two people are playing catch (or passing objects as they juggle) and the object they are throwing rises to a height of, say, nine feet as it travels between them, then the time the object is in the air is always constant - whether they are 3 feet apart or 30 feet apart.

The clues to discovery:
Dr. Crum and Dr. Lyons (both doctorates are honorary and self-bestowed) were experimenting with passing rings at a Saturday morning session of the Turks Head Jugglers. They noticed that, when one of them tried to pass a ring from behind their back, that the ring took an exceptional amount of time to arrive at the destination compared to a "normal" passing throw, seemingly independent of how hard it was thrown. This added time disrupted the synchronicity of the passing pattern. And they dropped things a lot.

With the razor sharp insight we have come to expect from these two giants of the physics of juggling (both legends in their own minds), they pounced on the realization that the behind-the-back throws had to be much higher than the normal throws - in order to clear the shoulder -- and the underlying physical principle stated above became immediately and abundantly clear.

The underlying physical principle described:
Anyone who ever survived a basic course in Physics will remember that the horizontal (X-axis) and vertical (Y-axis) components of the trajectory of a moving object are completely independent. Any change in one component has zero effect on the other.

Thus, if a ball is thrown by Person C at an upwards angle from a height of five feet and it rises to a height of nine feet, then arcs towards Person L, who catches it at a height of four feet, the duration of the flight can be completely determined by the formula distance equals one-half the acceleration constant times the time squared. The distance in this case is four feet (upwards travel) plus five feet (downwards travel).

But wait a minute!
Yes, of course, if the two persons are standing farther apart, the ball has to be thrown harder, but this is to compensate for the added horizontal separation only. The vector of the ball at time of release assumes a smaller angle to the horizon, since the horizontal component of the vector is longer, but the vertical component remains the same.

And no, air friction does not enter into the equation. If the two participants move farther apart, the thrower must take air friction into account when adjusting the throw and increase the horizontal velocity even more to accommodate the faster ball, but the effect of air friction on the vertical motion of the ball will remain the same.

So what?
Ah, the consequences of the C-LTOILD are both subtle, unobvious, and hard to describe. All at the same time.

One consequence is, as described above, you cannot throw an object harder and have it arrive sooner, if it has to start at a given point, rise to a second point, then and descend to a third point. So if you're passing rings, then all the throws must be as high as the behind-the-back throw, or you will get out of synch (unless it's thrown as a "double" -- picky, picky).

It also means that you can aim your throw at an imaginary point halfway between you and your juggling partner, and be sure of the timing - regardless of how far apart you may be standing.

Doctors Crum and Lyons await your comments on this remarkable insight. Click here to get to the Bulletin Board and tell them how smart you think they are.