of the Trajectories of Shot. 



279 



d 2 K 



Summary of Results ivhen n = 3, fx=. 



W 1000'* 



20 = a+/3 + ^— ^ ^— (ascending branch), . . , 



or 



or 



— fi~ a (descending branch), • . 



2^'=a+/3+ ^ ^ 3^ ( a ~" & (ascending branch), . . 



a + /3 — (/3 — a) (descending branch) ; . 



■v W,-. , -, 500,000/1000 1000\ 



X=^(l + cos2,)^(— . . . 



(b) 



(c) 



v W • o - 500,000/1000 1000\ 



m W /n . 250/1000 2 1000 2 \ 



T=_(l + cos2 )_(__ ~ ) 



• (e) 



These formulas might very easily be used if the calculator were furnished 

 with tables of and ^15^5^ where N ranges between the magni- 

 tudes of the velocity occurring in practice. 



Remarks on the Equation giving the Fall of Velocity in an Arc. 



§ 11. It will be observed that the two foregoing methods each open 

 with the same equation (a). Now there is a serious difficulty in the use 

 of that equation. Suppose, for example, we were to integrate over an arc 

 of 1° : we should have to use the mean value of K between its values 

 corresponding to the velocities at the beginning and end of the arc. But 

 we do not know the latter of these velocities ; it is the very thing 

 we have to find. The first steps in our work must therefore be to 

 guess at it. The practised calculator can, from his experience, make a 

 very good estimate. Having made his estimate, he determines K. He 

 uses this value of K in equation (a) ; and if he gets the velocity he 

 guessed at, he concludes that he guessed rightly and that he has got the 

 velocity at the end of the arc. If the equation (a) does not agree with 

 him, he makes another guess ; and so on, till he comes right. It seems 

 to me, however, that this method of going to work, leaving out of account 



