of the Trajectories of Shot. 275 



f-i 3g+(3n-7)(j>-g) 



_ x (3n-7)ff-(3n-10)ff 

 6 («-2)p-(»-3)2 * 



The value o£ X is now seen to become 



- {COS' 1 " 1 a + (n-l) C0S"~ 2 a sin a(a -/3)/}Q= - COS'^Q, 



where 



= a-(a-/3)/ 

 = ^+(i»/)(a»/3) 



2 ^ (n-2)p-(n-3)q' 6 



It is obvious that the value of T may be obtained in exactly the same 

 way, and that its magnitude is 



1 cos n-2 sin (j) Q. 



§ 7. It is necessary now to enter into an examination of the magnitudes 

 of the errors committed in neglecting squares of the quantity \p, and to 

 discuss how far (that is to say, for what size of arc) we may with safety 

 employ the values of X and T just obtained. 



Before doing so, I shall briefly remark on the step that we took in 

 neglecting (j>\) 2 in the value of /. On an examination of equation (9), it 

 will be seen that the principal part of the error committed amounts to 



12 



The error, therefore, in the angle (p is 



2n- 



12 q 



The consequent errors in X and Y 3 if we put D for the number of degrees 

 in a — (3, are given by 



, - 0-l)(2n-5) fp-q\* IT n 



(n-l)(2n-5)/j) 



X 



— = a similar expression. 



For the sake of simplicity, and to fix our ideas, let us take the case of 



u 2 



