20 



Mr. W. D. Niven on the Calculation 



[May 11, 



-^X = COS ^(S^sec^ — ^j^sec^)' • • (b) 



=^ Y = sill ^ (S^sec^-^/?sec^)' (c) 



^T = T ~ — T - (A\ 



"yy- g sec (P j?sec(9' (^Cl^ 



X, T being the horizontal and vertical distances respectively between A 

 and B, and T the time. 



The first equation gives q ; and it will in general be sufficient in that 



equation to put 0=""t^, because the secant o£ a small angle varies 



slowly. If, however, the angle of projection is large, it will be necessary 

 to operate twice with the equation (a), the first time to determine an 

 approximate value of g, the next time to determine a more accurate value 

 after having obtained an approximation to the correct value of 0. In 

 the (b) and (c) equations the more accurate value will be employed in 

 the cosine and sine, which occur as factors outside of the integrals. 



As an example of the method, take the case of a shot fired from a 

 38-ton gun. The following are the data: — Diameter of shot 12-5 

 inches, weight 810 lbs., angle of projection 3°, velocity 1400 feet, 

 height of muzzle above ground 14 feet. 



Let the work be first taken over the whole of the ascending branch. 

 The first thing to do is to find g from the formula (a). 



We have 



log sec lJ) = log (1400 cos 3° sec Ij) 

 = 3-1456812 ; 

 sec l| = 1398-6. 



From the Vj, Table, 



Y ^ =1-0568, 



¥ sec If 



Agam, 



d? - 

 log :^ = 1 •2853350 



log 3 = "4771213 

 log sec 1J= '0001488 



1-7626051 

 .-.^3 sec 1| =-5789; 

 V =1-0568-)- -5789. 



q sec If 



= 1-6357. 



.', from the Y Tables we find 



fl sec ] J = 1290. 



