6o 



SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 7I 



APPENDIX A 

 THEORY OF THE MOTION WITH DIRECT LIFT 

 Let M = the mass of the suspended system, comprising the chamber 

 together with any parts rigidly attached thereto, 

 mo = the mass of the expelled charge, comprising wadding and 

 the attached copper wire, the smokeless powder charge 

 (and also, in the experiments in vacuo, the black powder 

 priming charge), 

 V = the initial upward velocity of the mass M, 

 v = the average downward velocity of the mass m^ 

 and s = the upward displacement of the mass M. 

 We have at once for the initial velocity of the mass M, 



V^ = 2gs, 



and employing the Conservation of Momentum, we have for the 

 kinetic energy per gram of mass mg, expelled, 



v2 M2 



m. 



Sfs. 



APPENDIX B 



THEORY OF THE DISPLACExAiENTS FOR SIMPLE HARMONIC 



MOTION 



In addition to the notation given under Appendix A, the following 

 additional notation must be employed : 



Let ms=the mass of the spring, 



Fi = the force in dynes which produces unit extension of the 



spring, 

 m-i = the mass in dynes which produces unit extension of the 

 spring, 

 and s = the upward displacement of M, resulting from the firing, 

 that would be had if there were no friction. 



Then, allowing for the mass of the spring, we have, from the 

 theory of simple harmonic motion : 



Fx=(M+'|-)(f)'x. , 

 where x is any displacement, and P is the period of the motion. 



