Page:The Solar System - Six Lectures - Lowell.djvu/130



mean comet have successively aphelion distances from Jupiter's orbit to Saturn's.

The mean inclination we may take either as the mean of comets coming to us from all parts of space indifferently or as the mean of such parabolic comets as have actually been observed.

If we suppose the inclinations of the cometary orbits to be equally distributed through space, then the poles of the orbits will likewise be strewn uniformly over the celestial sphere. If α be the angle made by a pole with the pole of the ecliptic, the mean inclination of the poles can be found by multiplying the number of poles at any inclination, which is as the strip of surface yielding it, by that inclination, and then dividing the integral of this for the whole sphere by the surface of the sphere. The strip of surface at any inclination a is $$\scriptstyle 2\pi r^{2}\sin{\alpha}.d{\alpha}$$. Whence the average inclination in radians is

$$\scriptstyle \frac{\int^{\pi}_{0}{2\pi r^{2}\sin{\alpha}.\alpha.d\alpha}}{\int^{\pi}_{0}2\pi r^{2}\sin{\alpha}.d\alpha}=1$$

or i= 57°.3.

The second mean inclination or actual mean of all the parabolic orbits observed is i = 52°.4.