d'y

r

r

Acceleration of motion of the second at any instant; and, there being no fixed centre. velocity perpendicular to this plane at starting, there is there

fore none throughout the motion; thus the point moves in the
plane. And bad there been no acceleration, the point would
have described a straight line with uniform velocity, so that in
this case the areas described by the radius-vector would have
been proportional to the times. Also, the area actually described
in any instant depends on the length of the radius-vector and
the velocity perpendicular to it, and is shown below to be
unaffected by an acceleration parallel to the radius-vector.
Hence the second part of the proposition.
doc

dz
We have
P

=P
dto
dt*

de"
the fixed point being the origin, and P being some function of
2, y, z; in nature a function of r only.

donc Hence

at - y de

- 0, etc.,
which give on integration
dz dy

dx
y at

= C,, dt

-Y dt dt

dt dt Hence at once C,x+C2y + z = 0, or the motion is in a plane through the origin. Take this as the plane of xy, then we have only the one equation

dx

=C,= h (suppose).

dt
In polar co-ordinates this is

do dA
h =p 2

dz

dy

dac

=C

dy

- y dt

dt

dt

if A be the area intercepted by the curve, a fixed radius-vector, and the radius-vector of the moving point. Hence the area in

creases uniformly with the time. b. In the same case the velocity at any point is inversely as the perpendicular from the fixed point upon the tangent to the path, the momentary direction of motion.

For evidently the product of this perpendicular and the velocity gives double the area described in one second about the fixed point.

Or thus—if p be the perpendicular on the tangent,

dy dx P= x

- y

ds

ds dy dx and therefore

:h.
P

ข
dt dt dt

Acceleration directed to a fixed centre.

ds

=h.

If we refer the motion to co-ordinates in its own plane, we have ouly the equations

dx Px d'y_Py
dt

de

do whence, as before,

dt If, by the help of this last equation, we eliminate t from dx Pac

substituting polar for rectangular co-ordinates, we de arrive at the polar differential equation of the path. For variety, we may derive it from the formula of $ 32.

dor do\ They give

goe

=h. di

dt 1 Putting

(a) * = P,

= U, we have

u dt

+u =

1/dA Also = h'u, the substitution of which values gives us

du

Р d69 hu

(1), the equation required. The integral of this equation involves two arbitrary constants besides h, and the remaining constant belonging to the two differential equations of the second order above is to be introduced on the farther integration of

do
=hus

(2),
dt
when the value of u in terms of 0 is substituted from the equa-
tion of the path.

Other examples of these principles will be met with in the

chapters on Kinetics. Hodograph. 37. If from any fixed point, lines be drawn at every instant,

representing in magnitude and direction the velocity of a point describing any path in any manner, the extremities of these lines form a curve which is called the Hodograph. The invention of this construction is due to Sir W. R. Hamilton. One of the most beautiful of the many remarkable theorems to which it led him is that of g 38.

Since the radius-vector of the hodograph represents the velocity at each instant, it is evident ($ 27) that an elementary arc represents the velocity which must be compounded with the velocity at the beginning of the corresponding interval of time, to find the velocity at its end. Hence the velocity in the hodograph is equal to the acceleration in the patb; and the tangent to the hodograph is parallel to the direction of the acceleration in the path.

If x, y, z be the co-ordinates of the moving point, $, *, & those of the corresponding point of the hodograph, then evidently

dx dy
૬
n=

$

dt dt

de dn dg and therefore

dog dog

dz

dto dt dta
or the tangent to the hodograph is parallel to the acceleration in
the orbit. Also, if o be the arc of the hodograph,

do
dě

dζ
dt dt dt

dte

+

dt

+

C)* + C)

d

doz\?

dt dt dt2 or the velocity in the hodograph is equal to the rate of accelera

tion in the path. Hodograph

38. The hodograph for the motion of a planet or comet is comet, deo" always a circle, whatever be the form and dimensions of the orbit. Kepler's In the motion of a planet or comet, the acceleration is directed laws.

towards the sun's centre. Hence ($ 36, 6) the velocity is in

duced from

laws.

versely as the perpendicular from that point upon the tangent Hodograph
to the orbit. The orbit we assume to be a conic section, whose comet, de-
focus is the sun's centre. But we know that the intersection Kepler's
of the perpendicular with the tangent lies in the circle whose
diameter is the major axis, if the orbit be an ellipse or hyper-
bola; in the tangent at the vertex if a parabola. Measure off
on the perpendicular a third proportional to its own length and
any constant line; this portion will thus represent the velocity
in magnitude and in a direction perpendicular to its own-
so that the locus of the new points in each perpendicular will be
the hodograph turned through a right angle. But we see by
geometry* that the locus of these points is always a circle.
Hence the proposition. The hodograph surrounds its origin if
the orbit be an ellipse, passes through it if a parabola, and the
origin is without the hodograph if the orbit is a hyperbola.

For a projectile unresisted by the air, it will be shewn in
Kinetics that we have the equations (assumed in § 35, c)

d®

0,

dt de -9,
if the axis of y be taken vertically upwards.
Hence for the hodograph

de

0,
dt

9,
or = C, n=C'-gt, and the hodograph is a vertical straight
line along which the describing point moves uniformly.

For the case of a planet or comet, instead of assuming as Hodograph
above that the orbit is a conic with the sun in one focus, assume comet, de-
(Newton's deduction from that and the law of areas) that the
acceleration is in the direction of the radius-vector, and varies
inversely as the square of the distance. We have obviously

dx px d'y _ My
de got e di

dn dt

duced from Newton's law of force.

2003 ,

Hence for the hodograph

M2
($ + A)* + (n + B)' =

h?
the circle as before stated.
We may merely mention that the equation of the orbit will be

dac dy
found at once by eliminating and

dt

the three first dt

among integrals (1), (2), (3) above. We thus get

-h+ Ay - Bx =

tions of the

a conic section of which the origin is a focus.
Applica- 39. The intensity of heat and light emanating from a point,
hodograph. or from an uniformly radiating spherical surface, diminishes with

increasing distance according to the same law as gravitation.
Hence the amount of heat and light, which a planet receives
from the sun during any interval, is proportional to the time
integral of the acceleration during that interval, i.e. ($ 37) to
the corresponding arc of the hodograph. From this it is easy
to see, for example, that if a comet move in a parabola, the
amount of heat it receives from the sun in any interval is pro-
portional to the angle through which its direction of motion
turns during that interval. There is a corresponding theorem
for a planet moving in an ellipse, but somewhat more com-
plicated.

40. If two points move, each with a definite uniform velo-
city, one in a given curve, the other at every instant directing
its course towards the first describes a path which is called a

Curves of pursuit.