452 Mr. J. J. Sylvester ow Projectiles. 



Hence also 



ADC= 180-0- (90°- I) = 90°- |. 



Heuce 



2'' sin ACB 



A cos I 



and 



cos {l + (f))' 



<b , _ A sin A cos i 



?;cos^^ = AD= / ■. -. 



2 cos (t + 9) 



Hence eliminating t, we have 



v^ _ (sin<^)2 1 _ 1 — cos 



^Acost'~(l+ COS0) cos(t + (^) ~"cos(t + 0)* 

 If i = 0, i. e. if the gun is fired from the top of a battery com- 

 manding a level plain, we have simply 



sec .^ = 1 + 1^, 



which gives ^ the double of the angle of elevation. 

 In other cases we may make (j)-\-t, = -\^, we have then 



1— cos (-xlr — ^) 1 sin-vlr . v^ 



^-i Lz= ; i- smt— cosi= — rCOSt. 



cos Y" cos y- cos y' gn 



Let 



(l + |!) cot. = cote; 



then 



sin fc , , . , ■ 

 -. — cos (y — e) = l, 

 sme "^ 



sin e 

 cos (•»Ir— e)= -. — . 

 ^^ ' sm i 

 or 



tj, , X sine 



cos (<p + i—e)= -. — , 



^^ ' sm u 



from which ^, the double of the angle of elevation, may be de- 

 termined. 



Calling; -. — = cos 11, and taking 6., Ac, as the two values of 

 ° sm t u > • I ^ 



0, we have 20i + t— e=//., 



2</)2 + t-e = 360-ytt. 



01, 02 correspond to the angles of projection down and up the 

 slope respectively, the one affording what in an algebraical sense 

 is a maximum, and the other a minimum, but of course, arith- 



