434 PROFESSOR TAIT ON 



It therefore varies continuously from nil, at negative infinite values of s, to infinity at 

 positive infinite values. Any arc of the spiral ha3 therefore precisely the character 

 of the horizontal projection of the path of a sliced, toed, or heeled, golf-ball ; for it is 

 obvious at once that the curvature steadily increases with the diminishing speed of the 

 ball, thus far justifying the assumptions made in forming the equations of motion. We 

 have only to trace this spiral, once for all, to get the path for any circumstances of 

 projection. For the asymptote is obviously parallel to 



kwci 

 <p = tjt- — — a suppose. 



Measure $ from this direction, and the equation becomes 



i sla 



<p = ae • 



a gives the length corresponding to unit in the figure ; and a (which determines the point 

 of it from which the ball starts) depends only upon a and the ratio of the spin to the 

 initial speed. This, with <f>/a and s/a interchanged, is the equation of the equiangular 

 spiral, which would be the path if the resistance were directly as the speed. 



6. This enables us to get an approximate idea of the possible value of Jew in the flight 

 of a golf-ball. For if it be well sliced, its direction of motion when it reaches the ground 

 is often at right angles to the initial direction, although the whole deviation from a 

 straight path may not be more than 20 or 30 yards. Assume for a moment, what will 

 be fully justified later, that in such a case we may have (say) s= 480 feet, a= 240 feet, 

 and V = 350 foot-seconds. We see that 



so that 



^7 24 a a 



2 =»Xs;X6 , 4; 



^ w = o~^r = 0'357, nearly, 



o'o 



gives a sort of average value, which may safely be used in future calculations. In the 

 case just considered, the acceleration (at starting) due to the rotation, is 0*357 x 350 or nearly 

 four- fold that of gravity : i.e., the initial deflecting force is four times the weight of 

 the ball. 



7. In trying to find the positions of the asymptote, and of the pole, of the spiral of §5, 

 I spent a good deal of time on integrals like 



s'm(j>d<j> 



r 



Jo 



a + r/> 



with the hope of adapting them to easy numerical calculation by transformation to others 

 with finite limits, such as O — tt/2. Happily, I learned from Professor Chrystal that they 

 had been tabulated by Mr J. W. L. Glaisher; — and from his splendid paper (Phil. Trans., 

 1870) I obtained at once all that I sought. In fact his Si<£ and Ci<£ are simply the x,y 

 coordinates of this spiral (each divided by a) ; the axes being respectively the perpen- 

 dicular from the pole on the asymptote, and the asymptote itself. Thus I traced at once, 

 as shown in Fig. 1, the first three-quarters of a turn : — and the transformations I had 



