4&8 Dr. FIerschel’s Catalogue 
attraction of b on a = will be to the attraction of a on b 
— as n : in; which is again directly as ao : bo. 
I proceed now to explain a combination of three bodies, 
moving round a centre of hypothetical attraction. Fig. 5 con- 
tains a single orbit, wherein three equal bodies a b c, placed at 
equal distances, may revolve permanently. For, the real attrac- 
tion of b on a will be expressed by ; but this, reduced to the 
direction a 0, will be only • ; for, the attraction in the direc- 
tion ba is to that in the direction by , parallel to ao , as to 
The attraction also of c on a is equal to that of b on a ; 
therefore the whole attraction on a, in a direction towards 0, will 
be expressed by In the same manner we prove, that the 
attraction of a and c on b , in the direction bo, is ~ : and that 
of a and b on c, in the direction co, is . Hence, a b and 
c being equal, the attractions in the directions ao bo and co will 
also be equal ; and, consequently, in the direct ratio of these 
distances. Or rather, the hypothetical attractions being equal, it 
proves that, in order to revolve permanently, a b and c must be 
equal to each other. 
Instead of moving in one circular orbit, the three stars may 
revolve in three equal ellipses, round their common centre of 
gravity, as in Fig. 6. And here we should remark, that this 
centre of gravity will be situated in the common focus 0, of the 
three ellipses ; and that the absolute attraction towards that 
focus., will vary in the inverse ratio of the squares of the distances 
of any one of the stars from that centre, while the relative 
attractions remain in the direct ratio of their several distances 
