338 SOLUTION OF PROB. IX. B. I. OF THE PRINCIPIA. 



equal to the acceleration into dt produced by the central force 

 during the instant dt. Likewise PP' is the space described 

 with the initial velocity in the time dt. Compounding this with 

 the acceleration we have P' Q equal in magnitude and coincident 

 in direction with P'P". Therefore in the triangles RQP', 

 RP'P", we have, neglecting quantities of the third order, EQ 

 equal and parallel to R'P', QP' equal and in same direction 

 with P'P", and therefore the third side RP' equal and parallel 

 to R'P". That is to say, the point so moves as that its velocity 

 may always be resolved into two elements, both of them con- 

 stant; one, i.e. PR or P'R', normal to the radius vector, the 

 other, RP' or R'P", parallel to a fixed straight line. 



2. The velocity along the radius vector arises wholly from 

 the component RP', and varies as P'S. The velocity normal 

 to RP' arises wholly from the component RP and varies as 

 PT. Now these lines P'/S, PT, lie in similar triangles and are 

 as the hypotenuses, and therefore in a constant ratio. But as 

 the velocity towards the centre of force is proportional to the 

 velocity in a fixed direction, it is clear the point moves in a 

 conic section, because the fundamental property is, the distance 

 of any point from the focus is proportional to that from the 

 directrix. 



SCHOLIUM. It is obvious that if a boat rows at a given 

 rate through still water so that a line at right angles to the keel 

 always passes through a given point, the boat moves in a circle, 

 and the above demonstration shews that if the water is flowing 

 in a given direction with a constant velocity, the boat will move 

 as if attracted to the fixed point according to the natural law of 

 force, that is, it will describe a conic section with major axis 

 transverse to the stream. 



