1904.] Corrections to Naval Range-tables. 393 



To draw a similar curve for air-density increased in the ratio m : 1 

 proceed as follows : From the origin of co-ordinates draw straight 

 lines to points P, P, etc., on the range-table curve. In these lines take 

 points Q, Q', etc., so that OP/OQ = OP'/OQ' = etc. = m. Then the locus 

 of Q is the new curve. 



Stated verbally the rule is : To make a range-table for x per cent. 

 increase of air-density (or x per cent, increase of retardation due to 

 diameter and weight of shot), diminish each elevation and correspond- 

 ing range by x per cent. The elevations thus found are correct for 

 the ranges thus found with the new air-density. The time of flight 

 is diminished by x per cent. The remaining velocity found in the 

 range-table is the same for the new range-table at the altered range, 



Adopting the usual notation for exterior ballistics 



X is the range in feet ; V is the muzzle velocity. 

 C = w/nd*. 



w = weight of projectile ; d = diameter of projectile. 

 n = a constant depending on form of shot and air-density. 

 Vi = remaining velocity. 

 VQ = velocity at the vertex of trajectory. 

 t = time of flight ; < = elevation. 



Certain functions of velocity have been calculated and tabulated. 

 These are S (#), T (v), D (v) ; and the following are three well-known 

 equations : 



s<y)- ........................ (i)- 



From this we find v\ from the tables ; 



(2). 



Taking T (V), and T (?>i), from the tables, we know t. 



Now it is always assumed that for elevations up to 15 the vertex 

 of trajectory is reached in half of the time of flight. Hence 



whence we find V Q from the tables, which also give us the values D (V) 

 and D (VQ) in the third equation 



If the air-density be now increased in the ratio m to 1, and we use 

 letters with accents for the new conditions, 



mnd' 2 m 

 Let us find the elevation for a range X' where 



X' = X/m. Then X'/C' = X/C. 



2 G 2 



