THE PATH OF A .ROTATING SPHERICAL PROJECTILE. 505 



The corresponding value of y is about 27 feet. 



The point of contrary flexure is at e* /a = 2"03, so that x x = 255, and the value of -f 

 there has its maximum, about 0'07 only. 



In the other three paths above, the maximum ordinate and the maximum inclina- 

 tion both increase with the necessarily increased value of k, while the vertex and the 

 point of inflexion both occur earlier in the path. The approximate time of flight, in 

 all, is a little over five seconds. The paths themselves are shown, much foreshortened, 

 in figs. 10, 11, 12, 13, where the unit of the horizontal scale is 3*6 times that of the 

 vertical. This is given with the view of comparing and contrasting them. Fig. 14 

 shows the first, and flattest, of these paths in its proper form. It is clearly a fair 

 approximation to the actual facts ; and when we compare it with the others, as in the 

 foreshortened figures, we see that the assumption of constant spin (§ 4) is probably not 

 far from the truth. For, in the great majority of cases of drives of this character, there 

 is observed to be very little run : — and this can be accounted for only on the assump- 

 tion that there is considerable underspin left at the pitch. But it is also clear that the 

 falling off of the spin produces comparatively little increase of the obliquity of impact 

 on the ground, even in the exaggerated form in which these paths are drawn. Their 

 actual inclinations to the ground have tangents about 0*49, 0'66, 078, and 1*08 respec- 

 tively. The last, and greatest, of these angles is just over 45°. 



24. It is interesting to compare this set of data, and their consequences, with 

 those of §§ 11, 14, 15. The latter were in fair agreement with many of the more 

 easily observed features of a good drive, but they gave too high a trajectory. The new 

 measure of initial speed, and the consequent reduction of the estimated value of the 

 coefficient of resistance, have led to results more closely resembling the truth. 



But in all, as we have seen, there is one notable defect. The ball comes down too 

 obliquely, and this is the case more especially when the carry is a long one, and the ball's 

 speed therefore much reduced. I was at first inclined to attribute this to my having 

 assumed the spin to remain constant during the whole flight. This was my main reason 

 for carrying out the investigations described in §§ 22 sq. But these give little help, as 

 we have just seen, and I feel now convinced that the defect is due chiefly to the 

 assumption that the resistance is throughout proportional to the square of the speed. I 

 intend to construct an apparatus on the principle described in § 1 6 above, but of a much 

 lighter type, to measure the resistance for speed of 30 feet-seconds or so, downwards. 

 But I shall probably content myself with verifying, if I can, the idea just suggested ; 

 leaving to some one who has sufficient time at his disposal the working out of the details 

 when the resistance is proportional (towards the end of the path) to the speed directly, 

 or to a combination of this with the second power. The former is considerably more 

 troublesome than Robins' law ; and a combination of the two may probably be so labo- 

 rious as to damp the ardour of any but a genuine enthusiast. The possibility that the 

 law of resistance may change its form for low speeds (i.e., towards and beyond the 

 vertex of the path) throws some doubt upon the accuracy of the determination of the 



