440 PROF. TAIT ON THE PATH OF A ROTATING SPHERICAL PROJECTILE. 



so that the requisite speed is 548 foot-seconds; an increase of 56 per cent., involving 

 about 2*5 fold energ3>- of translation, which I take to be entirely beyond the power of 

 any player. And the time of flight is reduced to 3 s * 7 only, a rapidity of execution never 

 witnessed in so long a carry. The initial resistance in this case rises to nearly forty-fold 

 the weight of the ball. The equation of the path is 



2x\ 



v = 57-6 1-54 e - 1 ) 



9 a \ a) 



and the vertex is at 355, or about two-thirds of the range, only. 



15. Fig. 2 shows the three paths just described, which start initially in the same direc- 

 tion ; the uppermost is that with speed 350 and moderate spin. The lowest has the same 

 speed, but no spin. The intermediate course, also, has no spin, but the initial speed is 

 548 to enable it to have a range of 540 feet. Thus the two upper paths in this figure 

 are characteristic of the two modes of achieving a long carry : — viz. skill, and brute force, 

 respectively. In fig. 3 the first of these paths is repeated, and along with it are given 

 the corresponding trajectories with the same initial speed 350, but with inclinations of 

 0'12 and 0*0 respectively, and with the values of k, given above, which are required to 

 secure the same common range. [To increase this range from 180 to 250 yards, even in 

 the lowest and thus least advantageous path where there is no initial elevation, all that 

 is required is to raise the value of JcY (the initial acceleration due to rotation) from J 08 

 to 219 ; i.e. practically to double it. V might, perhaps, be increased by from 25 to 30 

 per cent, by a greatly increased effort in driving : — but k is much more easily increased. 

 A carry of 250 yards, in still air, is therefore quite compatible with our data, even if 

 there be no initial elevation. It can be achieved, for instance, if V is 400 foot-seconds, 

 and k about 50 per cent, greater than that which we have seen is given by a good slice. 

 Of course it will be easier of attainment if the true value of a is greater than 240 feet. 

 When there is no rotation there must be initial elevation ; and, even if we make it as 

 great as 1 in 4, the requisite speed of projection for a carry of 250 yards would be 1120 

 feet per second, or about that of sound.] Each of the curves has its vertex marked, 

 and also its point of inflexion, when it happens to possess one. Fig. 4 gives a rough, 

 conjectural, sketch of the probable form of the path if, other things being the same, the 

 spin could be very greatly increased. As I do not see an easy way to a moderately 

 approximate solution of this problem, either by calculation or by a graphic process, I 

 intend to attempt it experimentally. I am encouraged to persevere in this by the fact 

 that in one of the few trials which I have yet made, with a very weak bow, I managed to 

 make a golf-ball move point blank to a mark 30 yards off. When the string was adjusted 

 round the middle of the ball, instead of catching it lower, the droop in that distance was 

 usually about 8 feet. With a more powerful bow, and with one of the thin wooden shells 

 I have mentioned above, the circumstances will be very favourable for a path with a 

 kink in it. 



