of the Trajectories of Shot. 271 

 At the vertex of the trajectory the Telocity u e is given by 



I = A + ^p a ; 



u 11 «H 9 



If we put 



we have 



y = J<, (4) 



9 



(5) 



From equation (2), 



by (5), 



Again, 



'= 1 * — sec <pd(b 



J/3 



— sec 2 cty ; 

 _ ^ p sec 2 c7ft ^ (B) 



(It 



■J— J? 



Similarly, 



0-ri'?) 5 



sec 2 ^ cfy. ; 



(C) 



f <■ tan <f sec' ft cfy 



(l-yP f )= 



§ 4. The law of retardation found by Mr. Bashforth, in his own 

 symbols,, is expressed by 25 v 3 9 where 



AY (1000) 3 ' 



d being the diameter of the cross section of the shot in inches, "W its 

 weight in pounds, and K a quantity which is a function of the velocity, 

 and whose values are tabulated between certain values of the velocity for 

 every ten feet. In integrating the equations of motion over any arc, the 



