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.

(g) A circle about a point on its circumference; a parabola
about its vertex; about the intersection of axis and directrix.
(h) A conic section about a focus, 1/r — + ecosO.
(i) An ellipse or hyperbola about its center.
(j) An equilateral hyperbola about its center, (r/a)-2sec20..
2. A particle describes an ellipse, its acceleration being always perpendicular to one axis; find the law of acceleration.
3. What would be the path of a projectile if gravity varied
inversely as the cube of the distance from a horizontal plane?
4. If the acceleration varied as the distance in the preceding example?
5. Determine the path when the law is inversely as the square,
of the distance.
6. Consider the laws inversely as the cube of the distance,.
and inversely as the distance.
7. If the hodograph is a circle, and the field central, the law
must be inversely as the square of the distance.
8. The speed varies inversely as the distance from the center;.
determine the law of the field, and the path.
9. A planet is projected with the speed for. a circular path
at its given distance, but at an angle of 45 degrees to its radius
vector. Show that the point of projection is the end of a minor'
axis of its orbit, and its axes are as /2 to 1.
10. A particle, acted upon by two centers, moves with constant speed, and the product of its distances from the centers is
constant. If one center vary as the distance show that the other
varies as its inverse cube.
11. Show that the radius of the sphere of zero speed for a
planet is twice the major radius of its orbit, or that ~v2=-k/r —1/2a
12. Consider the motion of a particle in a field of two centers, each varying as the distance.
13. A particle moves in one plane, and its acceleration is
constant in magnitude, but revolves uniformly in direction. Find
the motion and path described, starting from rest.
Ans. With the x as the initial position and the x axis as the'
initial direction of acceleration; n, the angular speed of the latter,
and an2, its magnitude, the motion is x=aversnt, y=a (nt-sinnt),.
the motion of a point on a circumference of radius a, rolling unifornrly3 on the y axis.
14. A particle describes the cycloid x=averso, ya(-+sin4), in a fiield whose acceleration is always parallel to the base; determine its law.
Ans. The base is the line x —2a. If b is the constant ve —

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