THEORV AND PRACTICE

266

TRACT 37 .

part of it acts on tlic track, in order to bend it: whereas, iuthe horizontal or point-blank direction, the whole gravity isemployed to that purpose. This diminution will be nearlyas the cosine of the angle of elevation, being less and less asthat angle is greater, or is pointed nearer the perpendicular;so as that when pointed quite upright, it is not incurvatedat all, but the ball ascends perpendicularly upright.

PROBLEM xv.

To determine the Angle of Elevation of a Piece, to producethe greatest Range, also the Extent of that Range, with agiven projectile Velocity.

111. In the parabolic theory of projectiles, or those made

in empty or unresisting space, it is well known that, for thegreatest range, the elevation or direction of the piece, mustbe such as to bisect the angle contained between the vertexand the plane of the range ; that is, when the range is on thehorizontal plane, the direction of the pigee makes half a rightangle with the horizon, or with the vertical line; but whenthe range is on an ascending plane, or one making an angleabove the horizon, then the direction of the piece must makewith the vertical line, an angle less than half a right angle,by half the elevation of the plane above the horizon; andwhen the range is on a descending plane, or one making anangle below the horizon, then the direction of the piece mustmake with the vertical line, an angle greater than half aright angle, by half the angle of depression of the plane be-low the horizon. Also, in that theory, the trajoctory, or pathof the projectile, is a pcrlect parabola, luring the two partsof the curve, on both sides pt the axis, quite similar and equal,to. each other, ,

112. But, in projects made in the air, none of those beau-tiful properties take place; on the contrary, every thing is.irregular and perplexed ; the ascending .and descending