280 



Mr. W. D. Niven on the Calculation 



the loss of time, is open to objection in the point of accuracy. For, first, 

 there is no method of determining on what principle the mean value of 

 K is to be found — what manner of mean it is. Again, let us suppose 

 for an instant that the velocity at the end of the arc, guessed at, and the 

 value of K are in agreement ; that is to say, let the equation 



/1000V 3/3 /1000 V 3 ^ 2 K /T) -q v 

 (_-) -e 3 /3-(~-) sec3a= w _(P a -P,) 



hold for the values of vp and K used by the calculator. It by no means 

 follows that, he has hit on the right vp and K. For if he is dealing with 



a part of the tables in which happens to be nearly equal to 



_ 3W# sec 3 /3 (1000) 3 

 d 2 Pa-P/s 9 



it is obvious that there are ever so many pairs of values of vp and K 

 which will stand the test of satisfying the above equation. Now an exa- 

 mination of Mr. Bashforth's tables for ogival-headed shot shows that the 

 value of K diminishes as v increases from 1200 feet upwards, so that 



cCK. 



-r— is negative for a considerable range of yalues of v which are common 



dv ° 



in practice. It is not at all unlikely, therefore, that the value for 

 just stated may often be very nearly true ; in which case the 



process of guessing becomes extremely dangerous. 



I shall in the next method give a plan for determining the velocity at 

 the end of the arc, which seems to me simpler and more satisfactory than 

 the one we have been now discussing. Meanwhile, in connexion with 

 the present method, the following considerations are worthy of notice. 



§ 12. Let us consider the fundamental equation 



du d 2 K v 4 



d<p W g lOOOj 



If ^ were uniform, it would be the fall of horizontal velocity in 



change of inclination equal to the unit of circular measure. As that 

 angle is inconveniently large for discussions connected with trajectories 

 of shot, let us take 1° for unit-angle, and let D be the number of degrees 

 in the angle whose circular measure is 0. We then have 



W du 

 d 2 dD 



= 5_ZL_xl000x/'— V 



#180 \iooo/ 



The quantity on the right-hand side is the same for all shot of the same 

 kind ; and I propose to find its values for intervals of the values of - v. It 

 will be convenient to represent those values by the ordinates of a curve 



