Motion of Projectiles. 197 



sin (2 a e) = sin NEG, 

 and 



2a e = NEG^NEA + e; 

 consequently, 



-J- e. 



But because the angle NAM has its vertex in the circumfer- ^ 

 ence, and AM is a tangent, NAM is equal to JV72./2 ; also the 131. 

 angle MAP = e ; whence 



a = JVA/lf 4- ,/tf.tfP = NAP-, 



therefore the point N satisfies the question. 



The same may be shown with respect to the point N r . 

 deed in the triangle N'ELf, we have 



N'E : JV'L', or AE : El : : 1 : sin N'ELf, 



: : 1 ; sin N'EG, 

 whence 



AE sin N'EG = El; 

 and, since 



^2Esin (2a e) = E7, 

 as above shown, we have 



sin (2 a e) = sin N'EG, 

 and 



2 a e = JV'EG = JVi^ 4- e; 

 therefore, 



a = i JV^^f -f e = JV^JIf -f JIMP = 



309. Thus with the same force of projection, a projectile may 

 always be made to fall upon the same point M, according to 

 two different directions, provided that AP does not exceed DR. 

 The direction AN' is the most favourable for crushing buildings 

 or other objects with shells. The direction AN is to be pre- 

 ferred, when the purpose is simply to throw down walls and 

 breast-works, and by rebounding to lay waste at a distance. This 

 leads us to speak of ricochet firing ; but we shall first remark 

 that the equation <c = 1 1 cos a. found above, gives a simple ex- 



