Page:The Solar System - Six Lectures - Lowell.djvu/150



Then the probable amount of matter lying between $$x$$ and $$x+dx$$ is $$\textstyle \frac{h}{\sqrt{\pi}}e^{h^{2}x^{2}}dx$$, and considering $$x$$ to be $$y$$, and $$y$$, $$x$$, we have similarly for the probable amount of matter lying between $$y$$ and $$y$$ + $$dy$$, $$\textstyle \frac{h}{\sqrt{\pi}}e^{h^{2}y^{2}}dy$$

The probable amount, therefore, in the rectangle $$dx$$ $$dy$$ is $$\textstyle \frac{h^{2}}{\pi}e^{h^{2}(x^{2}+y{2})}dxdy=\frac{h^{2}}{\pi}e^{-h^{2}r^{2}}a$$ where $$a$$ $$dx$$ $$dy$$, and $$r$$ denotes its distance from the origin, or, in this case, the centre of the Sun.

For the amount in a ring at distance r, we have $$a=r dr$$.

Consequently it is evident that there is less relative variation in the density with the distance as one goes out. A fortiori, therefore, when the planetary masses do not increase in like proportion, the two ends, the outer and the inner, of the strip or bunch of matter that went to make each up, vary less in density inter se. In the resultant rotation, the speed of the separate particles counts for more, relatively, than their density, and, in consequence, for the outer planets we should get a retrograde rotation; for the inner, a direct one.

That the inner planets were not formed early in the system's development seems pointed at pretty