THE PATH OF A ROTATING SPHERICAL PROJECTILE. 503 



Effect of Gradual Diminution of Spin. 



22. In my former paper I assumed, throughout, that the spin of the ball remains 

 practically unchanged during the whole carry. That this is not far from the truth, is 

 pretty obvious from the latter part of the career of a sliced or a heeled ball. If, how- 

 ever, in accordance with § 4, we assume it also to fall off in a geometric ratio with the 

 space traversed : — an assumption which is probable rather than merely plausible ; so 

 long, at least, as we neglect the part of the loss which would occur even if the ball had 

 no translatory speed : — the equations of § 10 require but slight modification. For we 

 must now write, instead of h, 



The time rate at which this falls off is proportional to itself and to v, directly, and to b 

 inversely. 



If we confine ourselves to the very low trajectories which are now characteristic of 

 much of the best driving, we may neglect (as was provisionally done in § 10) the effect 

 of gravity on the speed of the ball, and write simply 



Thus the approximate equation of the path becomes 



dx~ a+ V {€ 1} 2V* {€ l) - 



Here 



— , = — ^ ; and finally 

 a' a b J 



y = ax + ^V /a ' ~ 1 " «/«') - f^ 2 (e 2r/a - 1 ~ 2x / a ) » 



where a is always very small, perhaps even negative ; and may, at least for our present 

 purpose, be neglected. Its main effect is to elevate, or depress, each point of the path 

 by an amount proportional to the distance from the origin ; and thus (when positive) 

 it enables us to obtain a given range with less underspin than would otherwise be 

 required. 



23. For calculation it is very convenient to begin by forming tables of values of the 

 functions 



AP) = —, and F{p)=- ^ = p ' 



for values of p at short intervals from to 3 or so. (Note that the same tables are 

 adaptable to negative values of p, since we have, obviously, 



X-P) = e- P f(p), and F(-p) = e -\f(p)-Fp)). 



