438 PROFESSOR TAIT ON 



the value of y will be slightly too small up to the point of inflection, and somewhat too 

 large up to (and some way beyond) the vertex of the path. 



1 1. When this paper was first read to the Society, it contained a considerable number of 

 details and sketches of the paths of golf-balls, based on three very different estimates of 

 the constant of resistance : — respectively much less than, nearly equal to, and considerably 

 greater than, that suggested by Bashforth's results. These details have just been printed 

 in Nature (June 29), and I therefore suppress them here, replacing them by calculations 

 based on experiments made between the two dates at the head of the paper. One 

 important remark, suggested by the appearance of these curves must, however, be made 

 now. Whatever, from 180 to 360 feet, be assumed as the value of a, the paths required 

 to give a range of 180 yards and a time of 6 s- 5, have a striking family resemblance. So 

 much do they agree in general form, that I do not think anything like an approximation 

 to the true value of a could be obtained from eye-observations alone. We must, therefore, 

 find a or V directly. Only the possession of a really trustworthy value of a, found by 

 such means, would justify the labour of attempting a closer approximation than that 

 given above. I have not as yet obtained the means of making any direct determinations 

 of a, but I have tried to find its value indirectly ; first, from experimental measures of V 

 made some years ago by means of a ballistic pendulum ; secondly, a few days ago, by 

 (what comes nearly to the same thing) measuring directly the speed of the club-head at 

 impact, and thus determining the speed from the known coefficient of restitution of the 

 ball. All of these experiments have been imperfect, mainly in consequence of the novelty 

 of the circumstances and the feeling of insecurity, or even of danger, which prevented the 

 player from doing his best. The results, however, seem to agree in showing that V is 

 somewhat over 300 foot-seconds (say, for trial, 350) for a really fine drive. Taking the 

 carry as 180 yards, and the time as 6 s , the value of a given by the formulas above is 

 somewhere about 240 feet. With these assumed data, the initial (direct) resistance to 

 the ball's motion is sixteen-fold its weight. Bashforth's results for iron spheres, when 

 we take account of the diameter and mass of a golf-ball, give about 280 feet as the value 

 of a. The difference (if it really exist) may possibly arise from the roughness of the 

 golf-ball, which we now see to be essential to long carry and to steady flight, inasmuch 

 as the ball is enabled by it to take readily a great amount of spin, and to avail itself of 

 that spin to the utmost. One of the arguments in §2 above would give the resistance 

 as proportional to y 2 + e 2 a> 2 , instead of to v 2 simply. 



12. We have thus all the data, except values of a and of k, required for the working out 

 of the details of the path by means of the approximate x, y equation just given. The 

 best course seems to be to assume values of a from 0"24 (according to Mr Hodge) down 

 to zero ; and to find for each the corresponding value of h which will make y = for 

 x = 540. This process gives the following values with a = 240, V = 350, as above : — 



a k kV/g alog. JcV/g 



024 



0182 



200 



166-3 



012 



0-246 



2-G9 



2375 



00 



0-309 



337 



2916 



