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. HODSON, WILLIAM STEPHEN RAIKES (1821-1858), known as "Hodson of Hodson's Horse," British leader of light cavalry during the Indian Mutiny, third son of the Rev. George Hodson, afterwards archdeacon of Stafford and canon of Lichfield, was born on the 19th of March 1821 at Maisemore Court, near Gloucester. He was educated at Rugby and Cambridge, and accepted a cadetship in the Indian army at the advanced age for those days of twenty-three. Joining the 2nd Bengal Grenadiers he went through the first Sikh War, and was present at the battles of Moodkee, Ferozeshah and Sobraon. In one of his letters home at this period he calls the campaign a "tissue of mismanagement, blunders, errors, ignorance and arrogance", and outspoken criticism such as this brought him many bitter enemies throughout his career, who made the most of undeniable faults of character. In 1847, through the influence of Sir Henry Lawrence, he was appointed adjutant of the corps of Guides, and in 1852 was promoted to the command of the Guides with the civil charge of Yusafzai. But his brusque and haughty demeanour to his equals made him many enemies. In 1855 two separate charges were brought against him. The first was that he had arbitrarily imprisoned a Pathan chief named Khadar Khan, on suspicion of being concerned in the murder of Colonel Mackeson. The man was acquitted, and Lord Dalhousie removed Hodson from his civil functions and remanded him to his regiment on account of his lack of judgment. The second charge was more serious, amounting to an accusation of malversation in the funds of his regiment. He was tried by a court of inquiry, who found that his conduct to natives had been "unjustifiable