62 ILLINOIS ACADEMY OF SCIENCE 
cess into single bodies these might not unlikely have retro- 
grade rotations ; that would depend however on the precise way 
in which the particles were brought together. But we need 
not dwell on this case for it is exceptional ; the Saturnian rings 
were developed under the conditions postulated by Roche 
and are a peculiar feature. Most orbits in the heavens are not 
circular and strictly concentric, as are these, but are elliptical, 
and the planetesimal orbits were by hypothesis notably ellip- 
tical. Now the velocity of a body in an inner elliptical orbit 
is indeed on the average higher than that of one in an outer 
elliptical orbit of the same type, just as in the case of circular 
orbits, but at the points where an inner elliptical orbit cuts an 
outer orbit, and where alone the bodies in these orbits can come 
together, the velocity of the body in the outer orbit is higher 
than that of the body in the inner orbit, precisely reversing the 
application of the law. This may be demonstrated mathemat- 
ically, but the layman may prefer to visualize it. This may be 
done in the simpler cases. If a notably smaller elliptical orbit 
is placed concentrically within a larger orbit of the same type, 
there can be no collision. It is only when the major axes are 
so moved that a more or less aphelion or distal portion of the 
smaller orbit is made to coincide with a more or less perihelion 
or proximate portion of the larger orbit that collision can oc- 
cur. If the dimensions be so selected that the precise aphelion 
point of the inner orbit can just touch the outer orbit at its 
perihelion point, it is easy to see that from this point the body 
in the inner orbit falls back in its onward course little by little 
toward the central body because its velocity is insufficient to 
maintain its aphelion distance. On the other hand, the body in 
the outer orbit moves steadily farther and farther from the 
central body in its onward course because its velocity is more 
than enough to maintain its perihelion distance, i. e. the com- 
mon distance of the two bodies when they were together. If 
the orbits appreciably cut one another, the inspection reveals 
similar relations of the two velocities but less clearly, and in 
other cases mathematical demonstration may be the only re- 
course. That will show, however, that this relation holds very 
generally but not universally. 
The actual rotatory effects of the union of two bodies in 
elliptical orbits vary with the precise conditions of their union. 
Keeping in mind that the body in the larger orbit moves the 
