128 On Machines in General*. 



be at the lowest point, i. e. the nearest possible from the fixed 

 point ; for this only happens in the case in which all the 

 parts of the system hold together and form but a single 

 body : on the contrary, in the case of natural gravity 3 it is 

 not necessary, in order that the centre of gravity should be 

 at the lowest point, that the parts of the system should be 

 united with each other. 



If bodies were attracted towards the fixed point in <the 

 inverse ratio of their distances to this point, the principle 

 alleged above would show that the /situation of equilibrium 

 is then the situation at which the sum of the products of 

 each mass, by the logarithm from its distance to the fixed 

 point, is a maximum. 



In general, if the bodies m of the system are attracted in 

 ratio of a power 77, from their distances x, to this point, 

 the situation of -equilibrium will be that at which the quan-r 

 tity s m x n + \ will be a maximum, or greater than in any 

 other situation ; i. e. when the difference of this quantity to 

 what it would be, if the system was in a situation infinitely 

 near, is equal to zero. 



If there are in the system several fixed points, towards 

 each of which the bodies m are attracted, in virtue of a 

 power given from their distances to this point; so that 

 #, 1/yZ, Sec. being the distances from m to these different 

 fixed points A x n , B y p , Cz'i, &c. are the central forces 

 of m towards these different foci, it will be the quantity 



ABC 



s m x n + 1 4- — ; s m y P + 1 + s m z 1 + * 



n+ l P+l J 9+ 1 



4- &c. which will be a maximum in the position of equili* 



brium. 



And if besides this, the bodies attract each other in ratio 



of any given power of the distances, in such a way that 



X expressing the distance from the molecule m to each 



of the other molecules of the system, F X r is the motrix 



force attractive of m towards this other molecule, the situa^- 



tion of equilibrium will be that where the quantity 



F A B 



• — smX. r + * + ■ 4jbx» + i+ smy P + * 



8f+3 n+l p+l 



C 

 + 



? + i 



