THE PATH OF A ROTATING SPHERICAL PROJECTILE. 433 



vw. I think too that the resistance in the line of flight will vary nearly as v 2 , irrespec- 

 tive of the velocity of rotation of the ball. 



As to the decrement of the energy of rotation, I think the second law which you 

 suggested is likely to be approximately true. The linear velocity due to rotation, even 

 at the surface where it is greatest, being supposed small, or at least tolerably small, 

 compared with the velocity of translation, I think you are right in saying that the 

 force acting laterally upon the ball will vary, at least approximately, as vw. If this 

 acted through the centre, it would have no moment. But I think it will not act through 

 the centre, though probably not far from it, so that it would have a moment varying as v<a. 

 Hence the decrement of angular velocity would vary as vw, and the decrement of energy 

 of rotation as w ( — dw/dt), or as w. vw, or as vw 2 , according to your second formula. 



However, I think the force at any point of the surface, of the nature of that which we 

 have been considering, would act very approximately towards the centre, and therefore 

 would have little moment, so that after all the moment of the force tending to check the 

 rotation may depend rather on the spin directly than on its combination with the velocity 

 of translation. But, if this be so, I doubt whether the diminution of rotation during the 

 short time that the ball is flying is sufficient to make it worth while to take it into 

 account." 



5. For a first enquiry, and one of great consequence as enabling us to get at least 

 general notions of the magnitude of the deflecting force, let us take the simple case of 

 a ball, projected in a direction perpendicular to its axis of rotation, in still air, and not 

 acted on by gravity. [This would be the case of a top or " pearie," with its axis vertical, 

 travelling on a smooth horizontal plane.] Suppose, further, that the rate of rotation is 

 constant. Then, in intrinsic coordinates, the equations of tangential and normal accelera- 

 tion given by our assumptions are 



s = — s 2 /a, and s 2 /p = s 2 -^ = Jews , 



respectively. The second may be put in either of the forms 



= Jew, or -j- = Jewjs . 



The first shows that the direction of motion revolves uniformly ; the second, that the 

 curvature is inversely as the speed of translation. And, as the first equation gives 



s = V i~ 8la > 



the intrinsic equation of the path is evidently 



if </> be measured from the initial direction of projection, and V be the initial speed. 

 This is an endless spiral, which has an asymptote, but no multiple points, and whose 

 curvature is 



Jew s i a 



V s * 



