38 S.MITirSDNIAN MISCELLANEOl-S COLLECTIONS VOL. 7I 



VALUES OF THE QUANTITIES OCCURRING IN THE EQUATIONS 

 The effective velocity, c(i — k). — The calculation which follows 

 has been carried out with the assumption '° of a velocity of ejection 

 of 7,500 ft./sec. and a constant, k, equal to yV- This velocity is con- 

 siderably less than those that were actually obtained, both in air and 

 in vacuo. The " effective velocity " will thus be 



c ( I — k) = 7,000 ft./sec. 



It should be noticed that k could be yV and yet not necessitate a 

 larger velocity of ejection than 7,640 ft./sec, which is also under the 

 highest velocities obtained in the experiments. It is important at this 

 point to remember that the velocities in vacuo would doubtless. have 

 been found to be considerably higher than the above value, if friction 

 could have been eliminated in the " direct-lift " method. 



The quantity, R. — The mean value of R for any interval is most 

 easily obtained from a graphical representation of P as a function 

 of V, the mean value of P between the beginning and end of the 

 interval being taken. Three curves have been used for this purpose : 

 for velocities ranging from zero to 1,000 ft./sec, 1,000 to 3,000 

 ft./sec, and from 3,000 ft./sec. upward. The first curve repre- 

 sented the experimental results of A. Frank ^ obtained with 

 prolate ellipsoids. The second curve represented the experimental 

 results of A. Mallock,^ whereas the third curve represented an 

 empirical formula by Mallock,^ which agrees well with experi- 

 mental results up to 4,500 ft./sec. — the highest velocity that has been 

 attained by projectiles — and hence may be used for still higher 

 velocities with a fair degree of safety. Mallock's expression, reduced 

 to the absolute ft. lb. sec. system and multiplied by \, the coefficient 

 for projectiles with pointed heads, becomes 



0.375 



P = o.oooo6432v2 /— j +480 (8) 



where v' = the velocity with which a wave is propagated in the air 

 immediately in front of the projectile ; which equals the 

 velocity of the body when that velocity exceeds the 

 velocity of sound in the undisturbed gas ; and 

 a=:the velocity of sound in the undisturbed gas. 

 The constant, 480 poundals, must be added for velocities over 2,400 

 ft./sec owing to the vacuum in the rear of the projectile. 



'A. Frank, Zeitschr. Verein Deutsches Ing. 50, pp, 593-612, 1906. 

 ' A. Mallock, Proc. Roy. Soc, 79A, pp. 262-273, 1907. 

 " A. Mallock, Proc. Roy. Soc, 79A, p. 267, 1907. 



