Royal Society. 235 



its position relatively to the least solid including them all ; and then 

 gives the equation between all the right lines drawn between n 

 points. 



Having shown that the result of differentiating the product of n 

 variables, m times successively may be derived from the mth power 

 of the sum of the n variables, developed by the polynomial theorem 

 by substituting for every power of each variable its differential of an 

 order numerically the same as the power ; and applied the theorem 

 to find the differential of the mth order of the equation between ten 

 right lines drawn between five points ; the author gives the first four 

 successive differentials of the same equation in another form. 



Proceeding with his investigation he deduces thfe necessary equa- 

 tion between the distances and central forces of five moving points, 

 and derives from it the general system of equations which determine 

 the motion of any number of spheres in terms of <p (the function of 

 the distance according to which the attractive force varies), their 

 masses and mutual distances. After proving that any number of 

 spheres may move so that the central force shall vary directly as the 

 distance, he shows that only certain values of <p are possible for an 

 infinite number of spheres, giving the criterion of possibility ; and 

 thence that the only possible law of central force for an infinite 

 number of spheres is that in which the force varies directly as the 

 distance. 



The author then enters upon some general considerations on the 

 physical impossibility of an universal law, rigorously exact and ex- 

 pressed by equations involving differentials of no higher order than 

 the second, and on the amount of disturbance by extraneous agen- 

 cies. Having shown how all equations expressed by rectangular 

 coordinates may be transformed into others involving only the mu- 

 tual distances of the spheres at m equal intervals of time, he gives 

 an equation of diflFerences defining the motion of n points, such that 

 the distances and their differentials of every order not exceeding m 

 may have any assigned values. 



After deducing a general formula for transforming equations of 

 differences not exceeding the mth order into equations between the 

 distances at m equal intervals of time, the author applies it to the 

 last equation, and shows that the equations so found are possible for 

 any number of moving points and for every value of m ; and that 

 the most general law, by which the motion of n equal spheres can 

 be determined, so that all move according to the same law at all 

 times, may be found by taking a proper value of m. He then shows 

 that these equations give a method of unlimited approximation to 

 any unknown law ; and suggests the mode of extending the solution 

 of the problem to solids of any figure and mass. Finally, he gives 

 the mth differential of the distance between any pair of points mo- 

 ving according to this law, in terms of the differentials of lower 

 orders including the distances. 



