of the Trajectories of Shot. 



287 



" The value of the mean angle (f to be employed in finding the time- 

 integral is 



f= a -ir+i <« -ft + < tJm ^)(«-« 2 > 



where the last term is the same as that in the above value of 0, but the 

 second term is only one half of its amount in the former case. 



" It will be seen that the above expressions for and <j>' are independent 

 of the value of n. 



" Also, if 



Q=l~ AO ~ 1) [(* ~ 2 ) (^c J - (* - 3)] (a-/3) 2 , 



where a — /3 is expressed as before in the circular measure, the values of 

 the coordinates X, T, and of the time T are given by 



T= K^2) Q(cos ^ r! shl ^(^-- i ^) =X(tan ^ ) ' 



"It may be remarked that if a and /3, as well as p and be interchanged, 

 the values of 0, 0', and Q will remain unaltered, and the values of X, T, 

 and T will merely change their signs, as it is evident should be the case. 



" The above values of 0, y, and Q are true to the third order of small 

 quantities inclusive, and the values of X, T, and T are true to the fourth 



order, considering EzS and a—/3 to be small quantities of the first 



order." 



