340 Mr. J. Prescott on the 



is not quite correct, that trajectory which is continuous at 

 the point of junction will be wrong for all values of the 

 velocity less than about 1300 feet per sec. We get much 

 better agreement with the true trajectory over large ranges 

 by getting two independent curves for the two laws of 

 resistance, and not troubling to join them up at the point 

 where u is 1060. This latter method is open to us when we 

 have a range table from which to determine our constants ; 

 but we should be obliged to make a continuous curve if we 

 had no other data than the initial velocity and the values of 

 I and Zj. 



18. By the same reasoning as that by which equation (14) 

 was proved, it can also be shown that, when u is less than 

 1060 feet per sec, the angle of departure, X , which gives 

 the range X, is given by 



2X 



Xtan\ =-Y'=-A-BX-C^. . . (21) 



We have already assumed that our trajectory is so low 

 that there is no appreciable difference between X and tanX . 

 Consequently, there will be no loss of accuracy in putting \ 

 for tan \ . 



19. It is known that a rifle jumps up or down on being 

 fired, so that the angle of elevation of the rifle barrel just 

 before firing, which is the angle observed in experiments, is 

 not the same thing as the angle of departure of the bullet ; 

 that is, the axis of the rifle before firing is not a true 

 tangent to the trajectory. If j denotes the upward jump of 

 the rifle, and 7 the angle of elevation of the rifle just before 

 firing, then \ = (y+j), or tan X = (7 + ;') approximately. 



20. If we now substitute (7+,;) for tan X in equations (14) 

 and (21), and then transfer X; to the right-hand sides of 

 these equations, we get a pair of equations which may be 

 written thus : 



X<y=p + qX + re kX when u> 1060, . . (22) 

 Xy=p' + q'X + r ! e k ' x when u< 1060, . . (23) 



where all the quantities except X and 7 are constants, and 

 wherein alsop= — r. In these equations we may assume, 

 for convenience, that the angle 7 is expressed in minutes 

 instead of in radians, since this only introduces a constant 

 factor all through the two equations. 



21. Let us write 5 for X7. Since a range table gives 

 corresponding values of X and 7, we can immediately 

 deduce the values of s therefrom. Now we shall denote by 



