[ahecht]

Square and eliptical gears are special because you can not only get dynamic ratios, but you can also get motions that you would normally need a linkage for.

Basically, a non-circular gear set works as follows:

If you have any two moving bodies, you can always define what are called "instant centers", which is a point that both bodies appear to rotate about in that instant. If you then plot the path that the instant center takes as the bodies move, you have a centrode.

Now, if you have a four bar linkage, you have two centrodes (since each centrode is defined by an instant center for two moving bodies). If you follow the logic though (which can be a bit complex without a further understanding of kinematics), you will see that if you make both centrodes into physical objects and roll them against each other, you will get the exact same motion and variations in angular velocity and mechanical advantage that you got from the four-bar linkage.

If your linkage had two fixed points (as most do), your non-circular gear set will also have two fixed points, which you can use to put a shaft through.

A side effect of all this is that, since circualr gears are just special cases of non-circular gears, that all gear systems are actually four-bar linkages in disguise. I know that this has probably not helped at all, and to understand it, you would probably need to take a course in kinematics. Perhaps someone else can explain it better...