( 491 ) 



XVI. — On the Path of a Rotating Spherical Projectile. II. 

 By Professor Tait. (With a Plate.) 



(Read 6th and 20th January 1896.) 



The first instalment of this paper was devoted in great part to the general subject 

 involved in its title, but many of the illustrations were derived from the special case of 

 the flight of a golf-ball. Since it was read I have endeavoured, alike by observation 

 and by experiment, to improve my numerical data for this interesting application, 

 particularly as regards the important question of the coefficient of resistance of the air. 

 As will be seen, I now find a value intermediate to those derived (by taking average 

 estimates of the mass and diameter of a golf-ball) from the results of Robins and of 

 Bashforth. This has been obtained indirectly by means of a considerable improve- 

 ment in the apparatus by which I had attempted to measure the initial speed of a golf- 

 ball. I have, still, little doubt that the speed may, occasionally, amount to the 300, 

 or perhaps even the 350, foot-seconds which I assumed provisionally in my former 

 paper : — but even the first of these is a somewhat extravagant estimate ; and I am now 

 of opinion that, even with very good driving, an initial speed of about 240 is not often 

 an underestimate, at least in careful play. From this, and the fact that six seconds 

 at least are required for a long carry (say 180 yards), I reckon the " terminal velocity " 

 at about 108, giving v 2 /S60 as the resistance-acceleration. 



I hope to recur to this question towards the end of the present paper : — but I should 

 repeat that I naturally preferred the comparatively recent determination to the much 

 older one, and that in formerly assuming a resistance even greater than that which 

 Bashporth's formula assigns, I was to some extent influenced by the consideration of 

 the important effects of roughening or hammering a golf-ball. For I fancied that this 

 might increase the direct resistance, as well as the effects due to rotation, by the better 

 grip of the air which it gives to the ball. [See last sentence of § 11. Of course the 

 assumption of increased coefficient of resistance required a corresponding increase of 

 the estimate of initial speed.] The time of describing 180 yards horizontally, i.e., when 

 gravity is not supposed to act, if the initial speed is 240 and the " terminal velocity " 

 108, is about 5 s *2 ; and this has to be increased by at least I s , if we allow for the 

 curvature of the path and the effect of gravity. I have employed this improved value 

 of the coefficient of resistance in all the calculations which have been made since I 

 obtained it. But various considerations have led me to the conclusion that the resist- 

 ance, towards the end of the path, may be somewhat underrated because of the assump - 

 tion that it is, throughout, proportional to the square of the speed. This point, also, 

 will be referred to later, as I wish to make at once all the necessary comments and 

 improvements on the part already published. 



VOL. XXXIX. PART II. (NO. 16). 4 E 



