α ( _v_ )² (5) ρ = 1.4 × --- × 10¹²(------) in suns per cu. LY, or β ( _d_α ) α ( _v_ )² = 2.8 × --- × 10⁻⁶(------) in atmospheres β ( _d_α ) α (6) _P_ = 9.15 _d_ --- _v_ The spirals form a continuous series from the great nebula of Andromeda to the limit of resolution, the smaller ones being much the more numerous. Considering them to be scattered at random as regards distance and size, some conception may be formed of their dimensions from the data at hand. The average radial velocity of those so far observed is about 400 km, while the proper motion is negligible. Putting the annual proper motion at 0.05″, the lower limit of the average distance is found to be about 7500 light-years. If they are within our sidereal system, then, as they are most numerous in the direction of its minor axis, the dimensions of our system must be much greater than is commonly supposed. The observations point to very large values for the rotational components of velocity, although the necessarily small scale of the instruments employed in their study renders the measuring difficult. Pease has determined the velocity of rotation for N.G.C. 4594 with some degree of accuracy. At 120″ from the nucleus it amounts to 300 km and varies linearly with the distance outward. V. M. Slipher reports that for the Andromeda nebula the angular rotation is fastest near the nucleus, and that this type of rotation promises to be the more common. Assume a typical spiral 400″ in diameter, with the ratio of the axes of figure as β/α = 0.1, and with rotation perpendicular to the line of sight at a velocity of, say, 200 km at the periphery. These figures are apparently not very different from the average of the two dozen brighter spirals. Table VII gives the dimensions in terms of distance. The velocity of escape is 280 km/sec. At the lower limit of average distance of spirals the typical nebulae would be fifteen light-years in diameter, forty-five million times as massive as our sun, and 3 × 10¹³ as dense as our atmosphere. On the other hand, if the typical spiral nebula is placed at a suitable distance, its dimensions assume the same order of magnitude as those of our own stellar system.