June 29, 1893J 



NATURE 



203 



of a sliced, or heeled, ball. And, by introducing an 

 acceleration in the plane of the path, constant in magni- 

 tude and direction, the path might be made to intersect 

 itself repeatedly. 



By the statement made above as to the whole change 

 of direction in the course of a well-sliced ball, and with 

 5' as the time of flight (for it, like the carry, is notably 

 reduced by slicing) we have 



2 



Thus it is clear that we may easily produce rotation 

 enough in a golf-ball to make the value oi ka as great as 

 o'3 or even o 4. And this can, of course, be greatly in- 

 creased when desired. This datum will be utilised later. 

 The fact (noticed above) that the time of flight, and the 

 carry, are both reduced by slicing, gives another illustra- 

 tion of the necessity for underspin when the time of 

 flight is to be long, and the carry far. 



Secondly : by a laboratory experiment which, I have 

 only recently learned, is due in principle to Robins. {Aji 

 Account 0/ Experiments relating to the Resistance of the 

 Air. R..S. 1747.) I suspended a wooden shell, turned 

 very thin, by a fine iron wire rigidly fixed in it, the other 

 end of the wire being similarly attached to the lower end 

 of a vertical spindle which could be made to rotate at 

 any desired rate by means of multiplying gear. Thin as 

 was the wire, it was but slightly twisted in any of the ex- 

 periments, so small was the moment of inertia of the 

 wooden shell. The wire acted as a universal flexure 

 joint ; and, by lengthening or shortening it I could make 

 the bairs mean speed, in small pendulum-oscillations, 

 vary within considerably wide limits. I verified this 

 result by substituting for the shell a leaden pellet of equal 

 mass but of far smaller radius, as I feared that some part 

 of the result might be due to stiffness of the wire, pro- 

 duced by torsion. But with the pellet the rotation of the 

 orbit was exceedingly slow. Thus u, and the average 

 value of s, could have any assigned values ; and from the 

 elli, tic form and the rate of rotation of the orbit of 

 the ball, the transverse force was found to be propor- 

 tional to either of them while the other was kept constant. 

 An exceedingly interesting class-illustration can be given 

 by making the ball revolve as a conical pendulum, and 

 while it is doing so giving it spin alternately with, and 

 opposite to, the direction of revolution. The effects on 

 the dimensions of the orbit and on the periodic time are 

 beautifully shown. This form of experiment could be 

 easily applied to considerable speeds, both of the transla- 

 tion and of rotation, if the use of a proper hall could be 

 secured. But it cannot be made strictly comparable 

 with the case of a golf-ball ; as the speed of translation 

 can never much exceed that for which the resistance is 

 as its first power only. [Robins' suspension was bifilar, and 

 the rotation he gave depended more on the twisting of the 

 two strings together than on the torsion of either. In this 

 mode of arrangement it is difficult to measure the rate of 

 spinning of the bob, and almost impossible to vary it at 

 pleasure.] 



We must next say a few words as to the manner in 

 which the spin, thus //-owrf to have so much influence on 

 the length of the carry, is usually given. I pointed out, 

 in the earliest article I wrote on the subject, " The 

 Unwritten Chapter on Golf" {Scotsman, Aug. 31, or 

 Nature, Sept. 22, 1887), that spin is necessarily pro- 

 duced when the direction of motion of the club-head, 

 as it strikes the ball, is not precisely perpendicular to the 

 face. Now, even when the head is not purposely laid a 

 little back in addressing the ball, (many of the longest 

 driversdo this without asking Why) it must always become 

 so in the act of striking if the player stand ever so little 

 behind the ball :— especially if, as Mr. Hutchinson so 

 strongly urges upon him, he makes the path of the head 

 at strikmg as nearly straight as possible. Mr. Hutchinson 

 gives a highly specious, but altogether fanciful, reason 

 NO. 1235. VOL. 48] 



for this advice. We now see why the suggestion is a 

 really valuable one. A " grassed " club, and especially a 

 spoon, gives this result more directly. As soon as I 

 recognised this, I saw that it furnished an explanation of 

 a fact which had long puzzled me : — viz. that one of my 

 friends used invariably to call for his short spoon when 

 he had to carry a bunker, so distant that it appeared im- 

 possible of negotiation by anything but a play-club. 

 And, if the ball be hit ever so little under the level of its 

 centre, with the upper edge of the face, very rapid under- 

 spin may be produced. This was probably at least one of 

 the objects aimed at (however unwittingly) by the best club 

 makers of last generation, for they made the faces of drivers 

 exception illy narrow. Some time ago I proposed, with 

 the same object in view, to bevel the face by deeply rasp- 

 ing off both its upper and lower edges : —thus in addition 

 saving the necessity for the "bone." 



I have neither leisure nor inclination to attempt (for 

 the present at least) more than a first approximation to 

 the form of the path under the conditions just pointed 

 I out. Anything further would involve a laborious process 

 of quadratures, mechanical or numerical, only to be 

 justified by the command of really accurate data as to the 

 values of a and V. I shall therefore at once assume that 

 neither gravity nor the spin affects the translatory speed 

 of the ball. (If the spin have such an effect, it will be 

 taken account of sufficiently by a slight change in the 

 constant of resistance ; and the effect of gravity on a low 

 trajectory is mainly to produce curvature which, in this 

 case, is to a great extent counteracted by the spin. It is 

 easy to see that the effects of this ignoration of gravity, in 

 the tangential equation of motion, are to make the path 

 rise a little too slowly at first, then too fast ; to make it 

 rise too high, and descend at too small a slope.) Hence 

 we may keep the first equation of motion above, and 

 write the second as 



<f 



s 



(tip 

 di' 



c . 



where tj) is reckoned positive in the ascending part of the 

 path ; and /t is written for/^u, its dimensions being those 

 of angular velocity. With the help of the value of s, 

 above, this becomes 



In A", ^ coordinates, ;r horizontal, this is nearly 



d.x' V V* ' 



Thus the x coordinate of the point of contrary flexure 

 is found from 



g ' 

 so that there must be such a point, i.e. the path is con- 

 cave upwards at starting, if ^V be ever so little greater 

 than g. 

 Again 



±-. ' 

 dx 



e + ^ W 



•)-^,(^^ 



2V'' 



I) 



where e is the initial slope. 



vertex is found by putting 



dy ^, 

 dx 



The X coordinate of the 



Finally, the approximate equation of the path is 



"vl 



y = ex + 



\^ -'-'-:)- §{--•-"} 



To deal expeditiously with these equations I formed a 

 table of values of the various factors in brackets, by the 

 help of Glaisher's data of natural antilogarithms [Camb. 

 Phil. Trans, xiii, 243). Next, I utilized in the equation 



1- 

 V/=: a(e« - I) 



