519. Since v = the greatest and least velocities are

p

ae, or=

at the nearer and farther vertices, or when p = a a+ac. Hence, if x be the upon the tangent when the velocity is an harmonic mean between the greatest and least velocities, we have

Also, if the force be considered constant the velocity acquired

521.

Let the area of the paper to be doubled down be

constantly equal to a2; then since the corner of the ▲ is a right

<, if the locus required be referred to the corner by the radius vector and angle originating with either edge of the paper, it may easily be proved that

and v2 = μ. (a2 — ç2)

a- being the space fallen through.

?

μędę

But if the force be considered constant, we have

where is the arc whose radius is a, and versine áp, as is easily shown on drawing the figure.

Hence if at the apse = R, the velocity of projection is

524.

Since the body is projected from an aspe, the velocity parallel to the plane will always be that of projection, which we call 8; then if y represent the to the plane, and ≈ the corresponding abscissa along it, we have

is a whole number, by assuming in the first case

To apply this equation to the case specified in the problem, we

(y

== √(-_-) × √(x − a) + C.

But x = 0, when ya; . C = 0; and

the equation to the common parabola, whose latus rectum is

B2 ordinate x and abscissa measured along the axis from the

-9

δμ

r being the radius of the circle, we have

2

T = √ F
√+ √ = = √F

Again, in the circle, the periodic time is (440)

T' = 2%