Rotation of these nebulae, as detected by the spectroscope, furnishes a means of relating mass and average density with distance. Assume an axis perpendicular to the line of sight, the mass as concentrated in the nucleus and the individual distant particles as rotating in equilibrium; let α be the radius, _P_ the period of rotation, _M_ the mass, and suppose the rotation to be circular. Then α³/_P²M_ = _C_, a constant. Let the unit of distance be the light-year (LY); of time, the year; of mass, that of the sun (S), then _C_, as computed from the earth-sun system, is about 4 × 10⁻¹⁵. Let α = the angular radius in seconds of arc _d_ = the distance in light-years ρ = the density in terms of earth’s atmosphere at sea-level _v_ = the linear velocity of rotation in km/sec. Then the following relations hold: (1) _M_ = 3.4 × 10⁻⁴ _d_α_v_² (2) ρ = 1.4 × 10⁻⁶ (_v_/_d_α)² (3) _P_ = 9.15 _d_α/_v_ The velocity of escape for the nebula is proportional to α/ρ and hence to _v_. This follows from the assumption that the particles are rotating in equilibrium, and therefore the factor of proportionality is the ratio between parabolic and circular velocity, that is, 1.4, and is independent of the distance. The value of _v_ for those nebulae so far observed is small, ranging from 5 to 10 km. Hence, if the assumptions held only approximately, the velocity of escape would be small and of the same order as that for the earth. Since these nebulae are composed of the lightest gases, it follows that at any save very low temperatures the molecules would escape at a very rapid rate. Certainly the nebulae would dissipate if the temperatures were of the order of that of our own atmosphere. TABLE VI =======+==========+=========+=================+================== _d_ | Diameter | Mass | Period | Density -------+----------+---------+-----------------+------------------ 10 LY | 0.001 LY | 1.2S | 1.5×10² year | 1.8×10⁻⁷ρ 10² | 0.01 | 12. | 1.5×10³ | 1.8×10⁻⁹ 10³ | 0.1 | 120. | 1.5×10⁴ | 1.8×10⁻¹¹ 10⁴ | 1.0 | 1200. | 1.5×10⁵ | 1.8×10⁻¹³ -------+----------+---------+-----------------+------------------ For an assumed typical planetary nebula, 20″ in diameter, rotating with a velocity of 6 km at 10″ from the perpendicular axis, Table VI has been constructed from formulae (1)-(3), expressing the order of magnitude of dimensions in terms of distance. The velocity of escape would be about 8.4 km per second, whatever the distance. Spectroscopic rotation of spirals furnishes an analogous set of formulae, and here the inclination of the axis may be roughly determined from the ratio of the two diameters of the nebulae. Let β be the semi-minor axis, then the formulae will be: (4) _M_ = 3.4 × 10⁻⁴ _d_α_v_²