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.

If OA=a, AC-b /c, the solution is
x=acos (c.t), y=bsin((Vc.t),
and for the path, (x/a))2+ (y/b) 2=1, the equation of an
ellipse referred to semi-conjugate diameters OA==a, OB=b.
In another plane through OA take OB' perpendicular to OA
as a new axis of y'; and consider the point P', whose coordinates are
x=acos(Vc.t),y'==asin(\/c.t). Show that as t varies P' describes
a circle AB', uniformly c radians per unit time, as P describes its
elliptic path AB, and that P'P is always parallel to the fixed line
B'B. The motion of the particle is therefore, the parallel projection of uniform circular motion whose period of revolution is
27r//c. This motion of P is called elliptic harmonic motion; it is
not, however, a harmonic motion on the ellipse. Harmonic motion
in a straight line is called simple harmonic motion.
28. A particle is projected from the origin with an initial
velocity whose components are a horizontal, and b vertical, and the
same component are u, v, at time t, when the particle is at (x, y).
The retardation of the air varies as the speed (its absolute value
being k), and gravity also acts. Find equations for u, v, x, y, in
terms of t. Also equations for the horizontal range and time of
flight, and the equations for maximum range, when the initial
speed is given.
RESULTANT MOTIONS
The general definition of the resultant of two motions is that,
if P. Q. be the moving points of the component motions at any time,
and 0 is a fixed point, then completing the parallelogram OPRQ,
R is the moving point of the resultant motion as to the origin 0.
Conceive a space carried by the moving point P, but with directions
unchanged; the point R is then carried with P in this space, and
has besides, the motion round P which Q has round 0. The motion
of Q round 0 is called the relative motion of R as to P, i. e.,
OQ=PR in length and direction at every instant of time.
The velocity and acceleration of the resultant point R above
are the resultants of the velocities and accelerations of P and Q,
since, as displacements, OR=OP+OQ, and hence for velocities,
D.OR=D.OP+D.OQ, etc.
EXAMPLES III
1. Show that a translation of the origin 0 to a new position.0' translates the resultant of two given motions the opposite
amount, 0'0.
2. Show that the parallel translation of a component motion.affects the resultant motion by the same translation.
3. The resultant of two simple harmonic motions of the

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