of the Trajectories of Shot. 



277 



Referring to the investigation of E in § 6, we see that the work for B' 

 will be the same, except that instead of a - /3 we should have to put an 

 angle i// corresponding to it, such that 



$ = a—j3 - ^t- tan a(a — /3) 2 . 



The error in X due to the difference between E' and E is /. 



-^T^COS^a sin 2 a(a-/3) 2 Q/. 



The integral S' may be reduced in the same way as the integral E. 

 The most important part of it will be found to be 



^ 3 



The corresponding error in X is therefore 



(2n sin 2 a - 1) cos"" 3 a Q. 

 ^yz 3 



In the former of these two components of the error of X it will be 

 sufficient for the purposes of an estimate to put/=|. In that case the 

 sum of the two component errors (call it SX) amounts to 



=~{ (n - 3} f' a - 2 } eos"-*«(a-^Q. 



It may be shown in a similar way that the error in T is given by 



,v 1 f(n— l)(n — 3) sin 2 a— 5n + ll"| , „_4 , ~ 2ri 



ci = --j ^ — i y sm a cos'^a(a— /3) Q. 



p I 12 J 



A discussion of these expressions for any assigned value of n would 

 determine for what magnitude of arc we might with safety employ the 

 formulae for X and Y. I shall confine myself to the case when n = 3, and 

 for the purposes of a ready estimate I shall take 



v 1 2 a + I) A 



X=-COS 2 — I^-Q, 



