342 PETER GUTHRIE TAIT 



defect, the too great obliquity of the descent, is due to the fact that, not knowing 

 the law according to which the spin falls off, I have assumed it to continue un- 

 changed throughout the path. The dotted curve, fig. 5, which gives a very close 

 approximation to the observed path, was obtained by rough calculations (little 

 more than estimates) from the same initial speed as fig. i, but with no elevation 

 to start with. The spin is initially about 50 per cent, greater than in fig. i, but it 

 has been assumed to fall off in geometric progression with the lapse of time. From 

 the mode in which this curve was obtained, I cannot insert on it, as I have done 

 on figs, i, 2 and 3, the points reached by the ball in each second of its flight; 

 but they will probably coincide pretty closely with those on fig. i. In the last- 

 mentioned figure, F is the point of contrary flexure and V the vertex. We have, 

 farther, 



Range ... ... ... ... ... 186 yards 



Time ... ... ... ... ... 6-2 



Greatest height ... ... ... ... 60 feet 



Position of vertex ... ... ... 0*71 of range 



In fig. 2 the initial speed and rotation are the same as in fig. i, but the 

 elevation is increased to 12. It will be seen that little additional carry is gained 

 in consequence. [Had there been no spin, the increase of elevation from 5 to 12 

 would have made a very large increase in the range.] In fig. 3 the elevation is 

 96 only, but the initial diminution of weight is treble of the weight. In this 

 figure we see well shown the effect of supposing the spin to be constant throughout, 

 for it has two points of contrary flexure, /*", and F t , and only between these is 

 it concave downwards. 



For contrast with these I have inserted, as fig. 4, a path with the same initial 

 speed, but without spin. Though it has the advantage of 15 of elevation, it is 

 obviously far inferior to any of the others in the transcendently important matter 

 of range. 



By comparing figs. 2 and 3, we see the effect of further increase of initial spin, 

 especially in the two points of contrary flexure in 3. Still further increasing the 



spin, these points of flexure close in upon the 

 vertex of the path, and, when they meet it, the 

 vertex becomes a cusp as in the second of the cuts 

 shown. The tangent at the cusp is vertical, and 

 the ball has no speed at that point. This is a 

 specially interesting case, the path of a gravitating 

 projectile nowhere concave downwards. With still further spin, the path has a kink, 

 as in the first of these figures. 



I have not yet been able to realise the kink (though I have reached the cusp 

 stage) with an ordinary golf ball. It would not be very difficult if we could get an 

 exceedingly hard ball, made hollow if necessary, and if we were to tee on a steepish 

 slope, and use a well-baffed cleek with a roughened face. I have, however, obtained 

 good kinks with other projectiles ; the first was one of the little French humming 



