498 PROFESSOR TAIT ON 



obtained from it by Mr Wood. I give the numerical data employed, plotting the 

 curves from a few of the calculated values of x and y. But I insert, at the side of each 

 trajectory, marks indicating the spaces passed over in successive seconds. This would 

 have been a work of great difficulty if we had adopted a direct process, even in cases 

 where the intrinsic equation can be obtained exactly : — and it must be carried out when 

 we desire to find the effects of wind upon the path of the ball. 



Fig.l represents the path when a = 240 (properly 234), F=300, <p Q = 0°, and k = 1/3. 

 This will be at once recognised as having a very close resemblance to the path of a 

 well-driven low ball. The vertex (at 0*76 of the range) and the point of contrary 

 flexure are indicated. This trajectory does not differ very much from that given (for 

 the same initial data) by the roughly approximate formula of § 10 ; which rises a little 

 higher, and has a range of some ten yards greater. But the assumed initial speed, and 

 consequently the coefficient of resistance, are both considerably too great. 



In fig. 2 all the initial data are the same except k, which is now increased to 1/2 : — i.e. 

 the spin is 50 per cent, greater than in fig.l. We see its effect mainly in the increased 

 height of the vertex, and in the introduction of a second point of contrary flexure. A 

 further increase of k will bring these points of contrary flexure nearer to one another, 

 till they finally meet in the vertex, which will then be a cusp, a point of momentary 

 rest, and the path throughout will be concave upwards ! This is one of the most curious 

 results of the investigation, and I have realized it with an ordinary golf-ball : — using a 

 cleek whose face made an angle of about 45° with the shaft and was furnished with 

 parallel triangular grooves, biting downwards, so as to ensure great underspin. [The 

 data for this case give extravagant results when employed in the formula of § 10. The 

 vertex it assigns is 510 feet from the starting point and at nearly 172 feet of elevation: 

 — while the range is increased by 60 or 70 yards. And that formula can never give 

 more than one point of contrary flexure. All this was, however, to be expected ; since 

 the formula was based on the express assumption that gravity has no direct effect on 

 the speed of the projectile.] 



Fig. 3 shows the result of dispensing altogether with initial rotation, while 

 endeavouring to compensate for its absence by giving an initial elevation of 15°. This 

 figure, also, will be recognised as characteristic of a well-known class of drives ; usually 

 produced when too high a tee is employed, and the player stands somewhat behind his 

 ball. Notice, particularly, how much the carry and the time of flight are reduced, 

 though the initial speed is the same. The slight underspin makes an extraordinary 

 difference, producing as it were an unbending of the path throughout its whole length, 

 and thus greatly increasing the portion above the horizon. But of course the pace of 

 the ball, when it reaches the ground, is very much greater than in the preceding cases, 

 it usually falls more obliquely, and it has no back-spin. On all these accounts we 

 should expect to find that the " run " will in general be very much greater. Still, in 

 consequence partly of the greater coefficient of resistance at low speeds, presently to be 

 discussed, over-spin (due to the disgraceful act called "topping") is indispensable lor 



