344.

Let x be the weight required. Then since P = Q,

345. force is P

g

2

=144 feet.

g

Let a denote the required weight. Then the moving

- X

Let F, F be the accelerating forces, then by the

F7

ť

F

Hence, if Q and Q' be the quantities of matter, the ratio of the moving forces is

348.

Let m be the degree of elasticity, and x the altitude required; also let a be the given velocity of projection. Then the velocity with which the body reaches the plane is √(a + 2gx),

and that with which it quits the plane is

But by the question x is the space due to this velocity, and the

349. Let a be the radius of the circle, (x, y); (x, y) the corresponding co-ordinates of the circle and locus measured from the horizon along the vertical diameter; then in any time t, considering the chord √x2 + y2 an inclined plane whose height is x', the form

12

944

gives xy"

also =—->, = 3/

x y’

|

350.

Let a, b, be the altitudes fallen by A, B, and x the space described along the horizon (which they will describe uniformly with the velocities acquired, since the plane of reflection pro

jects then along the horizon). Then, the velocities acquired being

if the whole time of motion be denoted by t, we have

Now, if a circle pass through the points of departure and touch the horizontal line, the distance, as is well known, of the point of contact from the concourse of the tangent and line passing through √ab... if a circle, &c. &c.

those points = √

351.

Q. E. D.

Let t, t', be the times down the chords corresponding to ordinates y, y', and abscissæ x, x'. Then considering the chords, planes inclined to the horizon by the

45° and 15°, we

y

tan. 15°

12+ √
√3
2-√3

353.

Let the distance of the bodies A, B, at first = a, and that of the required point C from Ax; then a, B, being the velocities with which the bodies impinge,

Now t being the time elapsed before the first impact of the bodies, we have