THE PATH OF A ROTATING SPHERICAL PROJECTILE. 493 



In the Beibldtter zu d. Ann. d. Phys. (1895, p. 289) there appears a somewhat 

 sarcastic notice of my former paper. The Reviewer, evidently annoyed at my remarks 

 on Magnus' treatment of Robins, which he is unable directly to controvert, refers to 

 Helie, Traite de Balistique, as containing an anticipation of my own work. I find 

 nothing there beyond a very small part of what was perfectly well known to Newton 

 and Robins ; except a few of the more immediately obvious mathematical consequences, 

 deduced from the hypothesis (for which no basis is assigned, save that it is the simplest 

 possible) that the transverse deflecting force due to rotation is proportional to the first 

 power of the translational speed. 



In the present article I give first a brief account of my recent attempts to determine 

 the initial speed of a golf-ball, and consequently to approximate to the coefficient of 

 v 2 in the assumed expression for the resistance. 



Next, instead of facing the labour of the second approximation (suggested in § 10) 

 to the solution of the differential equations, I have attempted by mere numerical calcu- 

 lation to take account of the effect of gravity on the speed of the projectile, and have 

 thus been enabled to give improved, though still rough, sketches of the form of the 

 trajectory when it is not excessively flat. This process furnishes, incidentally, the 

 means of finding the time of passage through any arc of the trajectory. 



Third, I treat of the effects of wind, regarded as a uniform horizontal translation of 

 the atmosphere parallel, or perpendicular, to the plane of the path. 



Finally, recurring to the limitation of a very flat trajectory, I have treated briefly 

 the effects of gradual diminution of spin during the flight. This loss is shown to be 

 inadequate to the explanation of the unexpectedly small inclination of the calculated 

 path when the projectile reaches the ground. Hence some other mode of accounting 

 for its nearly vertical fall is to be sought, and it is traced to the rapid diminution of the 

 resistance (assigned by Robins' law) when the speed has been greatly reduced. 



Determination of Initial Speed. 



16. The bob of my new ballistic pendulum was a stout metal tube, some 3 feet 

 long, suspended horizontally, near the floor, by two parallel pieces of clock-spring about 

 2*5 feet apart, and 8 '63 feet long. On one end of the tube was fixed transversely a 

 circular disc, 1 foot in diameter, covered with a thick layer of moist clay into which 

 the ball was driven from a distance of 4 feet or so. The whole bob had a mass of about 

 33 lbs. ; and, in the most favourable circumstances, its horizontal displacement was 

 about 3 "5 to 4 inches. As the ball's mass is 0*1 lb., the average indicated speed was 

 thus about 200 foot-seconds. * Though I had the assistance of two long drivers, whose 



* If l be the length (in feet) of the supporting straps, d the (small) horizontal deflection of the bob, its vertical rise 

 is obviously d 2 /2l, so that its utmost potential energy is 



(M+m)gd 2 l2l, 

 where M is its mass and m that of the ball. But, if V was the horizontal speed of the ball, that of bob and ball was 



