of 500 new Nebula, and Clusters of Stars. 487 
likewise be preserved in a permanent connection, by revolving 
in proper orbits about a common centre of motion. 
In all cases where stars are supposed to move round an empty 
centre, in equal periodical times, it may be proved that an ima- 
ginary attractive force may be supposed to be lodged in that 
centre, which increases in a direct ratio of the distances. For 
since, in different circles, by the law of centripetal forces, the 
squares of the periodical times are as the radii divided by the 
central attractive forces, it follows, that when these periodical 
times are equal, the forces will be as the radii. Hence we con- 
clude, that in any system of bodies, where the attractive forces 
of all the rest upon any one of them, when reduced to a direc- 
tion as coming from the empty centre, can be shewn to be in a 
direct ratio of the distance of that body from the centre, the 
system may revolve together without perturbation, and remain 
permanently connected without a central body. 
Hence may be proved, as has been mentioned before, that 
two stars will move round a hypothetical centre of attraction. 
For, let it be supposed that the empty centre 0, in Fig. 1 and 3, 
is possessed of an attractive force, increasing in the direct ratio 
of the distances oa : ob. Then, since here ao and bo are equal, 
the hypothetical attractions will be equal, and the bodies will 
revolve in equal times. That this agrees with the general law 
of attraction, is proved thus. The real attraction of b upon a is 
and that of a upon b is -^- r ; and, since b = a, it will be 
~r : ffr :: ao: bo ; which was required. 
In Figures 2 and 4, when the stars a and b are unequal, and 
their distances from 0 also unequal, let oa = n, and ob=zm; 
and let the mass of matter in a = m, and in b = n. Then the 
3 R 2 
