314 Revieio of the Principia of Newton. 



gental projection, and the curve is a circle. If the veloci- 

 ty of projection be still the same, but the direction of it be 

 not perpendicular to the radius vector, the differential of the 

 trajectory will have a constant ratio to the element of the mo- 

 tion perpendicular to the radius vector, and consequently the 

 curve will make equal angles with the radius vector. In both 

 cases the trajectory will make equal angles with the radius, 

 and is therefore the equi-angular spiral, for the circle is a 

 curve which every where makes equal angles with the radius. 

 If the velocity of projection be either more or less than 

 that which would be due to an infinite height, the differen- 

 tials of Newton's expressions become those of hyperbolic or 

 elliptical sectors, and of the distance of the centers of the 

 figures from the points of intersection of the tangent and 

 axis. The former figure will be used when the velocity is 

 less than that in the first cases, and the trajectory will be a 

 spiral constantly approaching the center of force ; but if the 

 velocity be greater than that due to an infinite height, the 

 curve by which the trajectory may be constructed, will be an 

 ellipse, and the trajectory itself a spiral constantly receding 

 from the center and terminating in an assymptote after an 

 infinite number of revolutions ; the degree of its approxima- 

 tion to the assymptote, will depend on the direction and ve- 

 locity of the moving body. When the trajectory becomes 

 the hyperbolie spiral, as under certain conditions it necessa- 

 rily must, this curve having its sub-tangent a constant quan- 

 tity, the centrifugal and centripetal forces will be equal, and 

 therefore no change in the paracentric velocity will arise from 

 the comparative effect of those causes, and the latter motion 

 will be uniform. This principle may not appear obvious, but 

 maybe shown thus: the equation of the hyperbolic spiral is 

 zw=a, where z is the radius vector, and -to the angle which it 

 makes with the axis of co-ordinates, whence w is always in- 

 versely as Z, and the centripetal or centrifugal force in cir- 



V 2 

 cles being always as ~n~ where V expresses the velocity in de- 

 scribing similar evanescent areas of circles by substituting 

 «- for V, we get F : ^T the centrifugal force = to the cen- 

 tripetal ; whence no variation of the paracentric velocity in 

 this curve can arise. Or thus, the absolute velocities in a 

 given time, or for a given area, will be inversely as the dis- 



