THE PATH OF A ROTATING SPHERICAL PROJECTILE. 437 



or the period in the rotating ellipse is always shortened : — whether the ball move round 

 it in the sense of the spin or not. This test cannot be applied with any certainty in the 

 experiment described above, for in general O is much less than n, so that m exceeds n 

 by a very small fraction only of its value. 



A very beautiful modification of this experiment consists in making the path of the 

 pendulum bob circular, before it is set in rotation. Then rotation, in the same sense as 

 the revolution, makes the orbit shrink and notably diminishes the period. Reverse the 

 rotation ; the orbit swells out, and the period becomes longer. 



1 0. The equations of motion of a golf-ball, which is rotating about an axis perpendicular 

 to its plane of flight, and moving in still air, are now easily seen to be 



s = ~--g S m<f>, 



<j> = k— ~COS(p . 



The most interesting case of this motion is a " long drive," as it is called, where (p is 

 always small, so long at least as it is positive ; its utmost average value for the first two- 

 thirds of the range being somewhere about 0*25. This applies up to, and about as much 

 beyond, the point of contrary flexure. A little after passing that point, <j> begins to 

 diminish at a considerably greater rate than that at which it had previously increased. 



A first approximation gives, as above, 



if we omit the term g sin <£ in the first equation. With this, the second equation gives 

 at once, on integration 



= a +y( 6 — 1 ) _ 2V I ~ '' 



We might substitute this for sin^> in the first equation, and so obtain a second, and now 

 very close, approximation to the value of s. But the result is far too cumbrous for con- 

 venient use in calculation. We will, therefore, be content for the present with the 

 rudely approximate value of s written above. 

 Integrating again with respect to s, we have 







Now, for rectangular coordinates (x horizontal) and the same origin, 



» =ycos<j)ds = /(l — ^-+&c.)ds, y=Jsin<pds =/(<£ — %+&c.)ds', 

 o o o o 



so that, to the order of approximation we have adopted, the equation of the path is 



y-ax+ y \i L-J W2 y & 1 a j. 



The only really serious defect in this approximation is the omission of gsincf) in the first 

 equation. This renders the value of s too large for the greater part of the path, and thus 



