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Mr. W. D. Niven on the Calculation 



^ Y — Sin (j) (S 2 g ec(? 5 — S paec( p), (c') 



__ T=T 2 g eC( p Tpg ec( p 



The quantities S„ and T v have been tabulated by Mr. Bashforth for a 

 considerable range of values of v, the upper limit being either 1700 

 or 2150, according as the shot is ogival-headed or spherical. The 

 tables for the ogival-headed shot have recently been revised and carried 

 to one place further in decimals. The quantities S c and T„ may 

 therefore be fortunately regarded as completely determined; and the 

 only question will be regarding the mean angle <p. Now it is a remarkable 

 circumstance that the value obtained for $ in § 7, if q is not widely different 

 from p, is very nearly the same for all values of n which are not far off from 

 3. But for limited portions of the trajectory the retardation may be con- 

 sidered as varying according to some simple power of the velocity, though 

 that power is not the same from point to point, but still not far from 3. 

 We may therefore take the value of found in § 7 as applicable to the 

 method now in hand. The adoption of this value of the mean angle, 

 since Y = Xtan0, is really equivalent to supposing the shot to move 

 parallel to the chord ; and the above proof shows what the limits of inte- 

 gration must be in order that the supposition may be made to approximate 

 to the actual case. The most sensitive quantity in this method, especially 

 near the vertex of the trajectory, is Y ; and in finding it over an arc cor- 

 responding to a change of inclination as large as 5°, it is necessary to use 

 the correct value obtained in § 7. As regards the other three quantities 

 given by the integrals (a'), (b'), and (d'), it will not matter much if we 



take ^-t^, at least for flat trajectories. If, however, the trajectory is not 



flat, and extreme accuracy is needful, it will be necessary to determine by 

 (a') the quantity q twice over — first an approximate value of it, in order 

 to get the mean angle 0, next its value with the mean angle used in the 

 equation. 



The following Table gives the value of" V„ for ogival-headed shot as low 

 as 900 feet per second. Below that value of the velocity Mr. Bashforth 

 does not give tables for the resistance, and the magnitudes of the resist- 

 ance for the lower velocities of ogival-headed shot have yet to be found. 



