Fig. 2. 



196 Captain Noble on the Ratio between the Forces tending to 



tion, it will be more convenient to consider the projectile in its 

 motion along the bore of the gun as moving on a fixed axis, and, 

 further, to suppose that the motion of rotation is communicated 

 to the projectile by a single groove. These suppositions will not 

 interfere with the accuracy of our results, and will enable us 

 very much to simplify the equations of motion. 



Take (fig. 2) as the plane of x y, the plane passing through the 

 commencement of the rifling at right angles to the axis of the 

 gun. Let the axis of x pass through the groove under conside- 

 ration, and let the axis of z be that of the gun. Let A P be the 

 helix, and let (see figs. 1 and 2) P(#, y, z) be the point at which 

 the resultant of all the pressures on the groove may be assumed 

 to act, the projectile being in a given position. Let the angle 

 AON = f 



Let us now consider the forces 

 which act upon the projectile. 

 We have, first, the gaseous pres- 

 sure acting on the base of the 

 shot. Let us call this force, the 

 resultant of which acts along the 

 axis of z, G. Secondly, if R be 

 the pressure between the projec- 

 tile and the groove at the point P, 

 this pressure will be exerted nor- 

 mally to the surface of the groove, 

 and if we denote by X, //,, v the 

 angles which the normal makes 

 with the coordinate axes, the re- 

 solved parts of this force will be 



R . cos X, R . cos fju, R . cos v. 



Thirdly, if fi Y be the coefficient of 

 friction between the rib of the 

 projectile and the driving-surface, 



the force /^R will tend to retard the motion of the projectile. 

 This force will act along the tangent to the helix which the point 

 P describes ; and if a, /3, 7 be the angles which the tangent makes 

 with the coordinate axes, we have as the resolved portions of this 

 force /AjR . cos a, /AjR . cos /?, /^R . cos 7 ; and summing up these 

 forces, we have the forces which act 



parallel to x=X =R|cosX— ^ cos a}, 

 parallel to y=Y =R|cosyLt— fi x cos/3L 

 parallel to z = Z =G + R|cos v— fi l cos7[ ; j 



and the equations of motion will be 



(i) 



