THE PATH OF A ROTATING SPHERICAL PROJECTILE. 439 



It will be seen that the values of h are of the order pointed to by the behaviour of a 

 sliced ball, though they are considerably less than that given in the example of §6. 

 This, of course, is a strong argument in favour of the present theory ; for, even in 

 the wildest of (unintentional) heeling, the face of the club is scarcely so much inclined to 

 its direction of motion as it is in good, ordinary, driving with a grassed club. (Slicing is 

 very much less susceptible of accurate quantitative estimation by means of eye-observa- 

 tions.) The third column gives the ratio of the initial deflecting force to the weight of 

 the ball. As this is more than unit in each of the three cases, all these paths are at first 

 concave upwards. The numbers in the fourth column indicate (in feet) the distance 

 along the range from the origin to the point of inflexion. 



The approximate equation of the first of these paths is 



y = 57.6 ^ + 30-05 (g*/ a _l-?)-3-76G 2 * /a -l-— ). 



The abscissa of the maximum ordinate is given by 



= 57-6 + 30-05(g :c / <l -l)-7-52(6 2 ^-l) 

 which leads to 



t x > a = 4'93, whence x = 384 nearly. 



The vertex is therefore at 0*71 of the range. 



13. Under exactly the same circumstances, had there been no rotation, the equation 

 of the path would have been 



y = 57-6- -3-76(V*/*-l--\ 



This gives for y = 0, 



x = 171 a = 410 feet only. 



The position of the vertex is given by 



= 57-6-7-52 (£ 2 */"-l); 

 so that 



x = 258 feet, nearly. 



In this case the vertex is at 0*63 of the range, only, and the time of flight is 3 s * 1. 



We have here, in consequence of a very moderate spin only, (in fact about half of 

 that given by a good slice), all other initial circumstances being the same, an exceedingly 

 well-marked difference in character between the two paths, as well as notable differences 

 in range, and time of flight. Thus, while a player who gives no spin has (say)' a carry of 

 136 yards only ; another, who gives the same initial speed and inclination of path but 

 also a very moderate amount of spin, accomplishes 180 yards with ease; his ball, in fact, 

 remaining twice as long in the air. 



14. For the sake of further illustration, let us consider the course by which the ball, sent 

 off at the same inclination, but without rotation, may be forced by mere initial speed to 

 have a range of 540 feet. Here the condition for V is 



= 129-6 -8^) 2 84-5, 



VOL. XXXVII. PART II. (NO. 21). 3 X 



