1 92 Dynamics. 



shall have = h, and consequently = 4 h ; 



therefore, 



4hx' = y' 2 . 



We hence infer that each point M of the curve AMC has this 

 property, that the square of the ordinate?/' or QM, parallel to the 

 tangent AZ, is equal to the product of the abscissa AQ or x' by 

 a constant quantity 4 h ; therefore the curve AMC is a parabola 

 which has for a diameter the vertical line AX, and for its param- 

 eter the quadruple of the height due to the velocity of projection, 

 and of which the angle AQM, made by the ordinates with this 



Trig, diameter, is the complement of the angle of projection ZAC. 



182. Tliis curve, therefore, is easily constructed, when the velocity of 

 projection and the angle of projection are known. 



304. We proceed to examine some of the properties of this 

 curve, considered as the path traced by a projectile ; and for 

 this purpose we refer the different points M to the horizontal line 

 AC by drawing PM perpendicular to AC. 



We designate AP by a?, PM by y, and the angle of projec- 

 Trig. 30. ti n ^^ kj a - I n tne right-angled triangle APNwe have 



1 : AN :: sin NAP : PJV, 

 :: cos NAP : AP-, 

 whence 



PN = AN sin NAP = -o t sin a, 

 and AP or x = v t cos a. 



Also, since MN = i g t 2 , as we have seen above, 



PM or y = v t sin a g t a . 

 Deducing from the former equation the value of tf, namely, 



and substituting it in the latter, we shall have, 



= x * ina . jg* 8 

 ** " cos a -y 2 cos a* ' 



