OT2

ot2, be drawn from the fixed point o parallel and equal to the velocities at p, p1, p2 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. [illustration] analytically thus (thomson and tait, _nat. phil._):--let x, y, z be the coordinates of p in the orbit, [xi], [eta], [zeta] those of the corresponding point t in the hodograph, then [xi] = dx/dt, [eta] = dy/dt, [zeta] = dz/dt; therefore d[xi] d[eta] d[zeta] ---------- = ----------- = ----------- (1). (d^2x/dt^2) (d^2y/dt^2) (d^2z/dt^2) also, if s be the arc of the hodograph, ds / /d[xi]\^2 /d[eta]\^2 /d[zeta]\^2 -- = v = / ( ----- ) + ( ------ ) + ( ------- ) dt \/ \ dt / \ dt / \ dt / / /d^2x\^2 /d^2y\^2 /d^2z\^2 = / ( --- ) + ( --- ) + ( --- ) (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.