Analytical dynamics, being a synopsis of leading topics in the analytical theory of dynamics with numerous examples and selections from Newton's Principia and other sources, by Arthur S. Hathaway.

13. In the preceding example, find the range on a plane of
depression 0 from the vertical and the maximum range.
Ans. r= —2a2sin (p —) sin p/gsin20.
For maximum range p —/20, r-a2/g(l+-cosO), the polar
equation of the curve which bounds the greatest distance that can
be reached in any direction with initial speed a, (a parabola with
the starting point as focus, axis downward, and highest point the
height of vertical flight). It is the envelope of the paths of all
particles starting from the origin with the same speed in the same
vertical plane.
15. A buys a flywheel of B 16 feet in diameter, with the
guarantee that it will run safely 80 revolutions per minute. The
wheel burst and a piece.was found at a distance of 1,000 feet. A
sues B for damages; should he recover?
16. Show that for a free path in a central field, with the
center as origin, the areal acceleration is zero, the areal velocity,
constant; and that consequently the free path of a particle in a
central field lies in a fixed plane containing the center, and the
radius vector from the center to the moving point describes equal
areas in equal times. Show also the converse, that if any free path
be a plane curve through a fixed point, and the radius from that
point to the particle describes equal areas in equal times, then the
field is central, the fixed point the center.
17. For a free path in any field show that the radial acceleration at time t is, in rectangular coordinates, D'x. (x/r) +
D2y. (y/r) +D2z. (/,r).
Note. The radial acceleration is the orthogonal projection of
the acceleration upon the radius vector OP, and is therefore the
sum of the orthogonal projections of its axial components. The
other component of acceleration, perpendicular to the radius vector, is called the transverse acceleration. Its component on the
x-axis is found by subtracting the x-component of radial acceleration from D2x, and is 2(Y"z-Z"y)/r2, where X", Y",Z", are the
components of areal acceleration.
18. Kepler announced, as the result of astronomical observations, covering many years, that the path of a planet is an
ellipse, with the sun at a focus; that the radius vector from the
sun to a planet describes equal areas in equal times; and that for
different planets, the cubes of the major radii of their orbits are as
the squares of their times of revolution. Newton thence proved:
(a) the acceleration on a planet forms a central field with the sun
as center; (b) towards the sun and inversely as the square of the
distance; (c) absolute acceleration the same for all planets.. Prove
the same in order, (a) from the plane path and law of areal speed,

23