202 



NA TURE 



[June 29, 1893 



the life-blood of their College to-day, the source of their 

 vitality, without whom they would have really little cause 

 for existence. 



Mr. Keeble, Natural Science Scholar of the College, 

 made a short and graceful reply. 



At the conclusion of dinner a move was made to the 

 Combination Room, where friendly and animated inter- 

 course was kept up for some time, and it was late before 

 the last of those engaged in the celebration separated for 

 the night. 



Breakfast was provided the following morning from 

 eight to ten for those resident in College overnight, and 

 by midday the guests had departed, leaving the courts 

 once more to solitude, and to their hosts a keen feeling 

 6f satisfaction at the honour done to the memory of 

 William Harvey and to the College by the recent presence 

 of so representative and distinguished a gathering of 

 visitors. 



SOME POINTS IN THE PHYSICS OF GOLF.^ 

 III. 



IN Part II of this paper (Natuire, Sept. 24, 1891) the 

 following statements were made :^ 

 " The only way ... of reconciling the results of calcul- 

 ation with the observed data is to assume that, for some 

 reason, the effects of gravity are at least partially counter- 

 acted. This, in still air, can only be a rotation due to 

 undercuttmg." 



" And, as a practical deduction from these principles, it 

 would appear that, to secure the longest possible carry, 

 the ball should be struck so as to take on considerable 

 spin ." 



: As these statements, and some of their consequences, 

 have been strenuously denied, I must once more show at 

 least the nature of the evidence for them 



, It depends, in one of its most telling forms, upon the 

 contrast' between the length of time a well-driven bail 

 remains in the air (as if in defiance of gravity) and the 

 comparatively paltry distance traversed. Everyone who 

 thinks at all on the subject must see that, without some 

 species of support, the ball could not pursue for six 

 seconds and a half a course of a mere 180 yards, nowhere 

 more than 100 feet above the ground. 



In fact, if we assume the initial slope of the path to be 

 I in 4, as determined for the average of fine drives by 

 Mr. Hodge with his clinometer (Nature, Aug. 28, 1890) 

 the carry of a non-rotating ball will be approximately (in 

 feet) 



where ^ is the acceleration due to gravity, T the time of 

 flight in seconds, and A a numerical quantity depending 

 on the resistance. The vahie of A varies continuously 

 between the limits, 2 for no resistance, and i for infinitely 

 great resistance. [It is assumed that the resistance is as 

 the square of the speed.] 



This formula gives, with the average observed value of 

 T (6^5, see Part II.) carries varying from about goo down 

 to 450 yards! The initial speed required varies from 

 416 foot-seconds upwards. The longest actually measured 

 carry on record, when there was no wind, is only 250 

 yards. Unfortunately, in that case T was not observed, 

 but analogy shows that it was probably much more than 

 7'. Even if we take it as 7= only, the " carry " ought to 

 have been, by the formula (which is based on the absence 

 of rotation), 522 yards at the very least ! 



I have purposely, in this example, kept to the case of 

 an initial slope of i in 4 ; because those (and they are 

 many, some of them excellent golfers) who altogether 

 reject the notion that undercutting lengthens the carry, 

 would of course in consistency refuse to believe that a 



1 Part of the substance of a paper on the Path of a Rotating Spherical 

 Projectile, read to the Royal Society of Edinburgh on June 5. 



NO. 1235, VOL. 48] 



long ball may sometimes start horizontally. But, to those 

 who allow this statement, the fact that the action of 

 gravity is occasionally largely interfered with, or i.-ven 

 counteracted, is obvious without any numerical calcula- 

 tions. In fact, from my present point of view, initial 

 slope is of little importance : — except, of course, in 

 avoiding hazards. The want of it is easily made up for 

 by a slightly increased rate of spin. 



Another way of looking at the matter is to assume, 

 from Mr. Hodge's data, 180 yards as a really fine carry, 

 and thence to calculate by the formula the requisite time 

 of flight. It varies from 4=1 to 2'''9 according as the re- 

 sistance, and therefore the necessary initial speed, are 

 gradually increased ; the former from nil to infinity, the 

 latter from 132 foot-seconds upwards. Thus the observed 

 time exceeds that which is really required when theie is 

 no spin, by 60 per cent, at the very least ! 



The necessity for underspin being thus demonstrated, 

 we have next to consider how its effect is to be introduced 

 in our equations. On this question I expressed a some- 

 what too despondent opinion in the previous part of this 

 paper. A rather perilous mode of argument (which 1 

 have since been able to make much more conclusive) first 

 suggested to me that the deflecting force, which is ler- 

 pendicular at once to the line of flight and to the axis of 

 rotation, must be at least approximately proportional to 

 the speed and the angular velocity conjointly. But I 

 tried (with some success) to verify this assumption by 

 various experimental processes. These, as will be seen, 

 led also to a numerical estimate of the magnitude of the 

 deflecting force. [And I was greatly encouraged in this 

 work by the opinion of Sir G. G. Stokes, who wrote : — " I 

 think your suggestion of the law of resistance a reason- 

 able one, and likely to be approximately true." This is 

 quite as much as I could have hoped for.] 



First : by the well-known phenomena called heeling, 

 toeing, and slicing, which are due to the ball's rotation 

 about a vertical axis. I have often seen a well-sliced 

 ball, after steadily skewing to the right through a carry of 

 150 yards or even less, finally move at right angles to its 

 initial direction, and retain very considerable spin when 

 it reached the ground. Neglecting the effects of giavity, 

 the equations of the path should be, in such a case. 



s = 

 P 



a 

 ksu> ; 



e.xpressing the accelerations in the tangent, and along 

 the radius of curvature, respectively. If we introduce 

 the inclination, <^, of the tangent to a fixed line in the 

 plane of the path, the second equation becomes 



(/> = ka>, 



showing tha: the time-rate of change of direction is pro- 

 portional to the speed of rotation. The first equation 

 gives, of course, 



s = Ve^a) 



where V is the initial speed. 



The space-rate of change of direction, i.e. the curvature 

 of the path, is thus 



d(p _ ka - 

 Is" V '"' 

 increasing in the same proportion as that in which the 

 speed of translation diminishes ; and, if we regard <■> as 

 practically unaltered during the short time of flight, the 

 intrinsic equation of the path is 



A rough tracing from this equation is easily seen to 

 reproduce distinctly all the characteristics of the motion 



