VOL. XLII.J PHILOSOPHICAL TRANSACTIONS. 645 



in a spiral that never arises to a given altitude. An example of each case is 

 given when the centripetal force is inversely as the fifth power of the dis- 

 tance. 



When the trajectory is described in a medium, let z be to a given magnitude 

 as the centripetal force is to the force by which the same trajectory could be 

 described in a void; and if the area be supposed to flow uniformly, the resist- 

 ance will be in the compound ratio of the fluxion of z, and of the fluxion of 

 the curve; and the density of the medium, supposing the resistance to be in 

 the compound ratio of the density and of the square of the velocity, shall be 

 as the fluxion of the logarithm of z directly, and the fluxion of the curve in- 

 versely. Hence, when any figure that can be described in a void by a force 

 that varies according to any power of the distance from the centre, is described 

 in a medium, the density of the medium must be inversely as the tangent of 

 the figure bounded by a perpendicular at the centre to the ray drawn from it to 

 the point of contact. 



After giving some properties of the trajectories that are described by a body 

 when it gravitates in right lines perpendicular to a given surface, and their ap- 

 plication to optical uses, the author proceeds to consider the motion of a body 

 that gravitates towards several centres. In such cases, that surface is said to be 

 horizontal, which is always perpendicular to the direction of the gravity that 

 results from the composition of the several forces; and it is shown, that the 

 velocity which is acquired by descending from one horizontal surface to another 

 is always the same, whether the body move in right lines, or in any curves; 

 the square of which is measured by the aggregate of several areas which have 

 the distances from the respective centres for their bases, and right lines propor- 

 tional to the forces at these distances for their ordinates. 



The force which acts on the moon is resolved into a force perpendicular to 

 the plane of the ecliptic, and a force parallel to it. This last is again resolved 

 into that which is parallel to the line of the syzigies, and that which is parallel 

 to the line joining the quadratures. The first measures the second fluxion of 

 the distance of the moon from that plane, the second and third measure the 

 second fluxions of her distances from the line of the quadratures, and from the 

 line of the syzigies, respectively. Hence a construction is derived of the tra- 

 jectory which would be described by the moon about the earth, in consequence 

 of their unequal gravitation towards the sun, if the gravity of the moon to- 

 wards the earth was as her distance from it. From this a computation is deduced 

 of the motion of the nodes of the moon, and of the variation of the inclina- 

 tion of the plane of her orbit. It is sufficient here to observe, that these mo- 



