496 PROFESSOR TAIT ON 



require some modification when k is such that (p may have large values, as for instance 

 in the kinked path treated below. But I do not pretend to treat the question exhaust- 

 ively, so that I merely allude to this source of imperfection of the investigation.] 



Let, now, a = 360, k = l/S, and suppose <p to be expressed in degrees. We have, to 

 a sufficient approximation, 



v '2 = ( v 2-4OOsin0)(l-^L), 



. 120 12000/, 1 \ 



and successive substitutions in these equations, starting from any assigned values of v 

 and <p, will give us the corresponding values for the next side of the polygon, with 

 the more recent estimate of the coefficient of resistance. See the two last examples in 

 §19 below, which lead to the trajectories figured as 5 and 6 in the Plate. 



Unfortunately, many of Mr Wood's calculations were finished before I had 

 arrived at my new estimate of the value of a ; but their results are all approximately 

 representative of possible trajectories : — the balls being regarded as a little larger, or a 

 little less dense, than an ordinary golf-ball ; in proportion as the coefficient of resistance 

 assumed is somewhat too great. And no difficulty arises from the assumption of too 

 great an initial speed ; for we may simply omit the early sides of the polygon, until we 

 come to a practically producible rate of motion. 



18. To discover how far this mode of approximation can be trusted, we have only 

 to compare its consequences with those of the exact solution. For the intrinsic equation 

 can easily be obtained in finite terms when there is no rotation. In fact, by elimination 

 of g between the differential equations of § 10, assuming Jc = 0, we have at once the 

 complete differential of the equation 



e sla v cos = Fcos <p = V suppose ; 



where it is to be particularly noticed that V is the speed of the horizontal component 

 of the velocity of projection, not the total speed. By means of this the second of the 

 equations becomes 



whence 



^(e»-l)=sec^ta„ 0o -sec^an^ + 1 og^±^. 



The following fragments show the nature and arrangement of the results in one of 

 the earlier of Mr Wood's calculated tables. Having assumed (for reasons stated in the 

 introductory remarks above) that a = 240, 1 supplied him with the following formulae :— 



v' 2 = (l-^y-4OOsin0(l-OO4), 

 0' = 0_12OOO COS 0(1 _o-04), 



