504 PROFESSOR TAIT ON 



These we will take for granted. We may now write 



y = ^ 2 (kVF(xla')-gF(2xja)) 



^i= «L(kVf(x/a')-gf(2xla)), 



The range, and the horizontal distances of the vertex and of the point of contrary 

 flexure, respectively, are given by the values of x which make the second factors vanish : 

 — and it is curious to remark that (to the present rough approximation, of course, and 

 for given values of a and a') these depend only upon the value of k V/g, i.e. the initial 

 ratio of the upward to the downward acceleration. Thus so far as the range is con- 

 cerned, the separate values of k and V are of no consequence, all depends on their 

 product. But it is quite otherwise as regards the flatness of the trajectory, for the 

 maximum height is inversely as the square of V. Of course we must remember that 

 one indispensable condition of the approximation with which we are dealing is that the 

 trajectory shall be very flat ; and thus, if the range is to be considerable, V cannot be 

 small, and (also of course) k cannot be very large. We have already seen how to obtain 

 a fairly approximate value of a (say 360), but b presents much greater difficulty. We 

 may, therefore, assume for it two moderate, and two extreme values, and compare the 

 characteristics of the resulting paths. If b be infinite, we have the case already treated, 

 in which the spin does not alter during the ball's flight ; while, if 6 be less than a, the 

 spin dies out faster than does the speed and we approximate (at least in the later part of 

 the path) to the case of no spin. Hence we may take for the values of b the following : — 

 oo, 900, 360, and 180 : — so that a' has the respective values 360, 600, oo , and -3G0. 

 Let the carry (x) be, once for all, taken as 180 yards. Then, for y = 0, we must have 

 2x/a = B; and the respective values of x/a' are 1*5, 0*9, 0, and — T5. With these 

 arguments the values of F are, in order, 



1-7873; 0-8807, 0-6908, 0-5, and 0-3258 ; 



so that we have the following approximate values of the ratio k V/g 



2-03, 2-59, 3-57, 5-49. 



The first two require a moderate amount of spin, only, if we take 240 as the initial 

 speed. 



The approximate position "of the vertex (x ) of the first of these paths is given by 



f(2xja) = 2-03f(x /a), or 6^ a = 3-06, (x /a = 11184) 

 whence x = 402*6, or about three-fourths of the carry. 



