326

miscellaneous

TRACT 38.

to the portion be of the parabola, bep, but turned the con-trary way. Conceiving either the two curves ae and ep,or the continued curve bep, to be described by a projectilein its motion, it is manifest that, whether the greater portionof the curve be described before or after the ball reaches thewall dr, will depend on its initial velocity, and on the dis-tance ac or uc, and on the angle of projection. 1 he pro-blem then is now reduced to this, viz, To find the angle atwhich a ball shall be projected from b, with a given impetus,so that the distance dp, at which it falls, from the given pointD, on the plane dp, parallel to the horizon, shall be a maxi-mum.

Now this problem may be P T. c

constructed in the following

manner: From any point Ein the horizontal line pc,

—

F ---

-- i

let fall the indefinite perp. A

L

i

eg, on which set off eb = the <!r

impetus corresponding to the given velocity, and bi = 2 \the distance of the horizontal plane below the point of pro-jection ; also, through I draw ap parallel to DC. From thepoint b set off bp = be + ei, and bisect the angle EBP bythe line bh : then will bh be the required direction of theball, and ip the maximum range on the plane ap.

For, since the ball moves from the point b, with the velo-city acquired by foiling through eb, it is manifest, from p.156 vol. 2 of the Math. Course, that DC is the directrix ofthe parabola described by the ball. And since both b and pare points in the curve, each of them must, from the natureof the parabola, be as far from the focus as it is from the di-rectrix ; therefore b and p will be the greatest distance fromeach other when the focus r is directly between them, thatis, when ep = BE + CP. And when bp is a maximum, sinceBi is constant, it is obvious that ip is a maximum also. Fur-ther, the angle Fbh being = ebh, the line bh is a tangentto the parabola at the point b, and consequently it is the dUrection necessary to give the range ip.