59] CALCULATING THE TRAJECTORIES OP SHOT. 481 



jtan y _ I . |^-| ( - ft) [^2 (sec y) 3 - T=T] 



1 __ 1 



^r (a /3) 2 tan y [ 2 w + 5 (sec y) 2 n 1 n + 3] >- ; 



( a - ) tan y - - ( a - )'"' C rT + 4 ( sec r)' 2 - + ] ; 



and these values are still true to the fourth order, considering and 



p + q 



a /3 to be small quantities of the first order as before. 



The angle which the chord of the arc makes with the axis of x becomes, 

 in like manner, 



which is true to the third order. 



The above expressions for X and Y may be transformed by introducing 

 this angle y into them instead of y, thus 



= (cos y)-> {l - ^ J~ (a - 0) tan y - ^ (a - 0)' (tan y) j . 



Hence we find 



X = 

 and 



= y-r- -^ f - r ,-~ _.,) (cosy)' 1 " 2 sin y 

 (n 2) \g" _p" 7 v 



{l - ^ ( a - W C^2 (sec y) 2 - ^3]} ; 



A. 



61 



