of the Trajectories of Shot. 



269 



The work is arranged under three heads, which are called the First, the 

 Second, and the Third Methods, intended to signify three distinct and 

 different solutions of the problem of finding the motion of shot. 



The First, which is the one adopted in Mr. Bashforth's treatise, is a 

 solution on the assumption that the retardation due to air-resistance is 

 fiv 3 , where ji is a constant. Now in the actual case fx is not constant ; 

 and therefore, in dealing with the equations of motion over any compo- 

 nent part of the trajectory, a mean value of li must be used. 



In contrast to the first method, the third adopts the mean value of 

 another quantity, viz. the inclination of the direction of motion. The 

 same thing was done in General Didion's solution, in which it was neces- 

 sary to employ the mean value of the cosine of the inclination. It will 

 be found that in selecting this mean we really implicate more quantities 

 than /i. But then there will be this advantage, that whereas there is no 

 way of determining the mean of li by the first method, and the greatest 

 uncertainty prevails regarding it when comparatively large arcs are inte- 

 grated over, according to the work now presented the nature of the mean 

 is investigated. In fact the determination of the mean angle may 

 be said to be the chief object and point in these investigations, because it 

 will be seen that there is no difficulty in establishing any of the equations 

 which will be used, and, excepting the labour, no difficulty in forming 

 the Tables. 



"What is described as the Second Method, although a distinct solution 

 when the retardation is formulated by fiv n , is to be regarded as a mere 

 stepping-stone to the one which follows it. It is in the second method 

 that all the quantities are expressed in terms of the mean inclination and 

 the magnitude of that angle determined. The quantity jx is taken con- 

 stant ; and therefore, in the case when n = 3, if we use the same mean 

 value of li as in the first method, we ought to get a solution much the 

 same. Beyond this, the second method possesses no further practical 

 value in the business now in hand, being entirely superseded in that 

 respect. Its chief value and importance consist in the determination of 

 the mean angle, because it is shown that the same value of that mean may 

 be used in the third method. 



This point being settled, it will be found that, according to the latter 

 method, all the required quantities will depend on three integrals, two of 

 which (the space- and time-integrals) have already been tabulated by Mr. 

 Bashforth. The third, which may be called the velocity-integral, is now 

 also tabulated for ogival-headed shot, and will be found further on. 



§ 2. It seems convenient at the outset to define the symbols which will 

 be employed throughout the work : — 



v will denote the velocity of the shot at any point of the trajectory ; 



u, the horizontal component of v ; 



0, the inclination of the direction of motion to the horizontal line in the 

 plane of the trajectory (the deflection from the plane being neglected) ; 



