Abstract

The relation between density distribution in a plane, circularly syrametrical disk, and the central force exerted upon points in its plane is investigated. For a disk in which the surface density can be expressed as a poiynomiai of the fifth degree in the distance from the center, it is possible to determine the force- curve in the plane, for interior and exterior points, by means of equations (8), (ii), and (i6), with the aid of Tables i and 2. The determination consists of the simultaneous solution of five linear equations. The curve of force can be converted into a curve of rotational velocity if simple circular motion is assumed. Conversely, it is possible to find the density distribution corresponding to a given force-curve. The observed rotational-velocity-curves of M 31 and M ~ are found to be adequately represented by thin-disk models. The solutions indicate a wide distribution of mass throughout both systems with but little tendency toward central concentration. The average projected density in the main body of M 3' is of the order of io3 O/psc2; and of M ~ 5 X 102 O/psca. With reasonable assumptions of the thickness of the two nebulae, the mean space density, averaged perpendicularly to the principal plane, is approxi- mately 2 O/psc3 in the main bodies of both objects. In the outer parts of M ~ the space density is prob- ably between o.o and 0.3 O/psc3, which is comparable with the density of matter in the solar neighbor- hood of the Galaxy. The total mass of M 31 is found to be 9.5 X ~Qb0 0; and of M ~ 1.7 X ~ 0. The constants of galactic rotation, in Oort's formulation of the problem, are re-examined with regard to the information they may yield concerning the distribution of mass. It is conc1uded~that the high cen- tral concentration of matter, commonly quoted for the Galaxy, is not a necessary consequence of the ob- servational data. I. INTRODUCTIO