THE PATH OF A ROTATING SPHERICAL PROJECTILE. 431 



closely does the position of the vertex agree with observation ; though it seems always 

 considerably too near the middle of the path. But the calculated time of flight, 

 which is greatest (for a given range) when there is no resistance, is always less 

 than two-thirds of that observed : — while, for high speeds, and correspondingly high 

 resistances, it is diminished to less than half the observed value. To make certain that 

 this discrepancy was not due to the want of approximation in my equations, yet without 

 the slightest hope of success in reconciling the various conflicting data, I made several 

 calculations by the help of Bashforth's very complete tables, which carry the approxima- 

 tion as far as could be wished ; but the state of matters seemed worse rather than better. 

 It then became clear to me that it is impossible for a projectile to pursue, for so long a 

 period as six seconds, a path of only 180 yards, no part of which is so much as 100 feet 

 above the ground : — unless there be some cause at work upon it which can, at least parti- 

 ally, counteract the effect of gravity. The only possible cause, in the circumstances, is 

 underspin : — and it must, therefore, necessarily characterise, to a greater or less degree, 

 every fine drive. (And I saw at once that I had not been mistaken in the opinion, which 

 I had long ago formed from observation and had frequently expressed, that the very longest 

 drives almost invariably go off at a comparatively slight elevation, and are concave 

 upwards for nearly half the range.) In Nature (24/9/91) I said : — 



" it thus appears that the rotation of the ball must play at least as essential 



a part in the grandest feature of the game, as it has long been known to do in those most 

 distressing peculiarities called heeling, toeing, slicing, &c." 



This conclusion, obvious as it seemed to myself, was vigorously contested by nearly 

 all of the more prominent golfers to whom I mentioned it : — being generally regarded as 

 a sort of accusation, implying that the best players were habitually guilty of something 

 quite as disgraceful as heeling or toeing, even though its effects might be beneficial 

 instead of disastrous. The physical cause of the underspin appears at once when we 

 consider that a good player usually tries to make the motion of the club-head as nearly 

 as possible horizontal when it strikes the ball from the tee, and that he stands a little 

 behind the tee. Thus the club-head is moving at impact in a direction not perpendicular 

 to the striking face ; and, unless the ball be at once perfectly spherical and perfectly 

 smooth, such treatment must give it underspin : — the more rapid the rougher are the 

 ball and the face of the club. This is, simply, Newton's "oblique racket." 



In fact, if the ball be treated as hard, and if the friction be sufficient to prevent 

 slipping, there is necessarily a maximum elevation (about 34°) producible by a club 

 moving horizontally at impact, however much "spooned" the face may be. This 

 maximum is produced when the face of the club makes, with the sole, an angle of about 

 28°: — which is less than that of the most exaggerated " baffy " I have seen. This, taken 

 along with the remark above (viz. that the longest drives usually go off at very small 

 elevations) is another independent proof that there is considerable underspin. 



Hence the practical conclusion, that the face of a spoon, if it is to do its proper work 

 efficiently, ought to be as smooth as possible. 



