306 Royal Society. 



resistance at 1300 feet per second is only 9*94 lbs. per square inch, 

 whilst, according to that hypothesis, the back resistance alone would 

 be 15 lbs. per square inch. 



It is suggested that the true reason of the great increase of resist- 

 ance may be found in the fact that a wave-impulse cannot be pro- 

 pagated at a greater velocity than 1100 feet per second, and that 

 consequently a great condensation of air must take place in front of 

 the projectile at all velocities exceeding this, and the resisting force 

 of such condensed air will increase at a greater rate than indicated 

 by Mariotte's law, owing to the evolution of heat due to the con- 

 densation. 



A comparison is then instituted between the resistances as ascer- 

 tained by the above law and those given by Hutton's formula. 



It is stated that in experiments made on May 17th, 1867, the 

 small shot weighing 8*8 lbs., moving with a mean velocity of 986 feet 

 per second, lost 58^ feet of velocity in a distance of 900 feet. 



The time of flight being *96 of a second, the resisting force must 

 have been nearly twice the weight of the shot, or more accurately 

 17*2 lbs. 



Now, according to the formula given in this paper, the resist- 

 ance is found to be 17*75 lbs., whilst Hutton's formula gives a 

 resistance of 46| lbs. 



Having thus obtained a law which gives, with considerable accu- 

 racy, the residual velocity at any point of the flight, the correspond- 

 ing equation to the trajectory is deduced for low degrees of elevation 

 when the length of the arc differs very slightly from the horizontal 

 distance, or ds = dx nearly ; and the following is the resulting equa- 

 tion : — 



r- 2(m+I) n+2 2(^+1) "| 



v=a :tan <f+ A|^ IJ « » +a - .-—^ („+«) » J- 



where A=-^-~. — — , and cand a are the constants, and n the index 

 2 _ n + 2 

 O 

 in the general equation (z-\-a)v n =C. 



Examples of the application of this are given, showing the calculated 

 elevation for the 12-pounder muzzle-loading Armstrong gun for 

 ranges of 2855 yards and 4719 yards, the gun being 17 feet above 

 the planes. 



The calculated elevations were 6° 56' and 14° 6', the actual ele- 

 vations being 7° and 15° respectively. 



It is not intended to claim more than approximate accuracy for 

 the formulae in this paper. The general formula has been shown to 

 be derived by taking mean values of n and c, whereas the actual re- 

 sults would indicate that the value of n increases with the diameter 

 of the projectile ; and it is shown in a note that the values of n which 

 agree best with experiment are, 



for the small shot ?z==2'4, 

 for the large shot n=4, 



