# 1911 Encyclopædia Britannica/Hodograph

**HODOGRAPH** (Gr. ὁδός, a way, and γράφειν, to write), a curve of which the radius vector is proportional to the velocity of a
moving particle. It appears to have been used by James
Bradley, but for its practical development we are mainly indebted
to Sir William Rowan Hamilton, who published an account of it
in the *Proceedings of the Royal Irish Academy*, 1846. If a point
be in motion in any orbit and with any velocity, and if, at each
instant, a line be drawn from a
fixed point parallel and equal to
the velocity of the moving point at that instant, the extremities
of these lines will lie on a curve called the hodograph. Let PP_{1}P_{2}
be the path of the moving point, and let OT, OT_{1}, OT_{2}, be drawn
from the fixed point O parallel
and equal to the velocities at
P, P_{1}, P_{2} respectively, then the
locus of T is the hodograph of the
orbits described by P (see figure).
From this definition we have
the following important fundamental
property which belongs
to all hodographs, viz. that at
any point the tangent to the
hodograph is parallel to the
direction, and the velocity in
the hodograph equal to the
magnitude of the resultant
acceleration at the corresponding
point of the orbit. This
will be evident if we consider
that, since radii vectores of the
hodograph represent velocities in the orbit, the elementary
arc between two consecutive radii vectores of the hodograph
represents the velocity which must be compounded
with the velocity of the moving point at the beginning of any
short interval of time to get the velocity at the end of that
interval, that is to say, represents the change of velocity for
that interval. Hence the elementary arc divided by the element of
time is the rate of change of velocity of the moving-point, or in
other words, the velocity in the hodograph is the acceleration in
the orbit.

Analytically thus (Thomson and Tait, *Nat. Phil.*):—Let *x*, *y*, *z*
be the coordinates of P in the orbit, ξ, η, ζ those of the corresponding
point T in the hodograph, then

ξ = | dx |
, η = | dy |
, ζ = | dz |
; |

dt | dt | dt |

therefore

dξ |
= | dη |
= | dζ |

d ^{2}x/dt ^{2} | d ^{2}y/dt ^{2} |
d ^{2}z/dt ^{2} |

Also, if *s* be the arc of the hodograph,

_{_____________________________________} | ||||||||||||

ds |
= v = √ ( | dξ |
) | ^{2} |
+ ( | dη |
) | ^{2} |
+ ( | dζ |
) | ^{2} |

dt | dt |
dt |
dt |

_{______________________________________} | |||||||||||

= √( | d ^{2}x |
) | ^{2} |
+ ( | d ^{2}y |
) | ^{2} |
+ ( | d ^{2}z |
) | ^{2} |

dt ^{2} |
dt ^{2} |
dt ^{2} |

Equation (1) shows that the tangent to the hodograph is parallel to the line of resultant acceleration, and (2) that the velocity in the hodograph is equal to the acceleration.

Every orbit must clearly have a hodograph, and, conversely, every hodograph a corresponding orbit; and, theoretically speaking, it is possible to deduce the one from the other, having given the other circumstances of the motion.

For applications of the hodograph to the solution of kinematical problems see Mechanics.