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 |
| d²x/dt² | d²y/dt² | d²z/dt² |
(1).
Also, if s be the arc of the hodograph,
| ds | = v = √ [( | dξ | ) | ² | + ( | dη | ) | ² | + ( | dζ | ) | ² | ] |
| dt | dt | dt | dt |
| = √ [( | d²x | ) | ² | + ( | d²y | ) | ² | + ( | d²z | ) | ² | ] |
| dt² | dt² | dt² |
(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.