644 fHILOSOPHICAL TRANSACTIONS. [aNNO 1742-3. 



logarithmic spiral when m = 3, an epicyloid when m = 4, a circle that passes 

 through the centre of the forces when m = 5, and the lemniscata when m-=.'] . 

 In general, it is constructed by drawinjg a perpendicular from the centre of the 

 forces to a right line given in position, and any other ray to the same right line, 

 then increasing or diminishing the angle contained by this ray and the perpen- 

 dicular in the given ratio of 2 to the difference between 3 and m, and increasing 

 or diminishing the logarithm of the ray in the same given ratio. The trajec- 

 tories described in analogous cases by centrifugal forces, are constructed in a 

 similar manner. These are the figures in which the perpendicular, from a given 

 centre on the tangent, is always as some power of the ray drawn from the same 

 centre to the point of contact, which are afterwards found to arise in the reso- 

 lution of the most simple cases of problems of various kinds. 



When the area described about the centre of an ellipse is given, the subtense 

 of the angle of contact, drawn through one extremity of the arc parallel to the 

 semidiameter drawn to the other extremity, is in a given ratio to this semidia- 

 meter; and therefore, when an ellipse is described by a force directed towards 

 the centre, that force is always as the distance from the centre. When the 

 force is directed toward the focus, it is inversely as the square of the distance. 

 And these two cases are considered particularly, because of their usefulness in 

 the true theory of gravity. To illustrate which, the laws of centripetal forces 

 that would cause a body to descend continually toward the centre, or ascend 

 from it, are distinguished from those which cause the body to approach towards 

 the centre, and recede from it by turns. A body approaches from the higher 

 apsid toward the centre, when its velocity is less than what is requisite to carry 

 it in a circle; and if its velocity increase, while it descends, in a higher pro- 

 portion than the velocities requisite to carry bodies in circles about the same 

 centre, the "velocity in the lower part of the curve may exceed the velocity in a 

 circle at the same distance, and thereby become sufficient to carry off the body 

 again. But while the distance decreases, if the velocities in circles increase 

 in the same, or in a higher proportion, than the velocity in a trajectory can in- 

 crease, the body must either continually approach toward the centre, if it once 

 begin to approach to it, or recede continually from the centre, if it once begin 

 to ascend from it; and this is the case, when the centripetal force increases as 

 the cube of the distance decreases, or in a higher proportion. But though, in 

 such cases, the body approaches continually towards the centre, we are not to 

 conclude, that it will always approach to it till it fall into it, or come within 

 any given distance ; for it is demonstrated afterwards, in art. 879 and 880, that 

 it may approach to the centre for ever, in a spiral that never descends to a 

 given circle described in the same plane, and that it may recede from it for ever 



