of the Trajectories of Shot. 287 



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

 integral is 



^=^+if^f(«-/3) + i(tan^)(a-^, 



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

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



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

 of the value of n. 



"Also, if 



Q-l-A(«-l)[(n-2)-(geo^) ~(«-3)](«-/V, 



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

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



fi(n-l) \q n f 7 



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

 the values of (p, </>', and Q will remain unaltered, and the values of X, Y, 

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



" The above values of <f>, <p' 9 and Q are true to the third order of small 

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



order, considering 2—2 and a — ft to be small quantities of the first 



order/' 



VOL. XXVI. 



x 



