362 Mr. J. Prescott on the 



59. We have been assuming throughout the preceding 

 argument that m 2 T 2 >rf. If this inequality is reversed, then 

 the quantity we have called s changes its form. Thus 



»'-ft*r, 



^"— * T 'dT, 



-w 



= is/:rf{ log e T - log e ( ^rf+ */rf-m 2 T 2 ) } + i ^rf-m?H\ 



Therefore e~ xs contains a factor T^ rf , which is such a large 

 power of T that it increases very rapidly as T increases. 

 However small <£ may be initially ? therefore, it very quickly 

 becomes great, and consequently (z — i/r) becomes great. 

 But for a bullet travelling at a great speed, the angle t/r, 

 which measures the angular deflexion of the line of flight, 

 will not change so quickly as z } which is the cause of i/r. 

 That is, z must be large before yfr can become large. It 

 follows, then, that the shot will be stable travelling nose fore- 

 most provided m 2 T 2 >r/, but unstable if this inequality is 

 reversed. 



60. We will consider what sort of motion is represented 

 by equation (73) when the axis is stable. It would be a 

 considerable feat to integrate this equation to get i/r. After 

 all we do not really need to get the precise motion. It will 

 be sufficient to know the type of the motion. It is safe to 

 conclude that i|r will be similar to ty in that it will contain 

 a factor e imT , and that its terms will also contain factors e is 

 and e~ is . Likewise z will be similar to both in these respects. 

 Consequently all the four quantities e, tj, sc, y will consist of 

 such terms as Pcos (F-fmT±s), where P is a function of T, 

 and F is a constant. Now m is such a large number that 

 (mT + s) increases very rapidly, making the periodic terms 

 pass very quickly through their successive maxima and 

 minima. Then each of the two terms in yjr which have 

 coefficients Ex and E 2 indicates a very rapid conical motion 

 of the axis, and a consequent helical motion of the shot in 

 its path. The angles of the cone are not constant, and the 

 helices do not lie on a circular cylinder. In the complete 

 motion of the axis the two conical motions are superposed ; 

 that is, the axis of the shot describes one cone about a line 

 which itself describes the other cone. The angle of one of 

 the cones, the one represented by the term containing the 

 factor e~ i \ is a rapidly diminishing angle. 



