HODOGRAPH (Gr. &56s, a way, and ypagav, to write)

This article appears in Volume V13, Page 558 of the Encyclopedia Britannica.

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HODOGRAPH (Gr. &56s, a way, and ypagav, 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

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

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 PPIP2 be the path of the moving point, and let OT, OTI, OT2, be

drawn

DRAWN

from the fixed point 0 parallel and equal to the velocities at P, PI, 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 he 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

CHANGE (derived through the Fr. from the Late Lat. cambium, cambiare, to barter; the ultimate derivation is probably from the root which appears in the Gr. 冠 urmu', to bend)

of velocity for that interval. Hence the elementary arc divided by the

element

ELEMENT (Lat. elementum)

of time is the rate of

change

CHANGE (derived through the Fr. from the Late Lat. cambium, cambiare, to barter; the ultimate derivation is probably from the root which appears in the Gr. 冠 urmu', to bend)

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, , n, i- those of the corresponding point T in the hodograph, then
dx o=dt, 1"=di;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

TISSUE (Fr. tissu, tissue, participle of tisser, Lat. texere, to weave)

of mismanagement, blunders, errors, ignorance and arrogance ", and outspoken criticism such as this brought him many bitter enemies throughout his career, IN ho 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 18J5 two separate charges were brought against him. The first was that he had arbitrarily imprisoned a Pathan

chief

CHIEF (from Fr. chef, head, Lat. caput)

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 and oppressive," that he had used abusive language to his native officers and personal violence to his men, and that his system of accounts was " calculated to screen peculation and fraud." Subsequently another inquiry was carried out by Major Reynell'Taylor, which dealt simply with Hodson's accounts and found them to be " an honest and correct record . . . irregularly kept." At this time the Guides were split up into numerous detachments, and there was a system of advances which made the accounts very complicated. The verdicts of the two inquiries may be set against each other, and this particular charge declared " not proven." It is possible that Hodson was careless and extravagant in money matters rather than actually dishonest; but there were several similar charges against him. During a tour through Kashmir with Sir Henry Lawrence he kept the purse and Sir Henry could never obtain an account from him; subsequently Sir George Lawrence accused him of embezzling the funds of the Lawrence Asylum at Kasauli; while Sir Neville Chamberlain in a published

letter

LETTER (through Fr. lettre from Lat. littera or litera, letter of the alphabet; the origin of the Latin word is obscure; it has probably no connexion with the root of linere, to smear, i.e. with wax, for an inscription with a stilus)

says of the third brother, Lord Lawrence, " I am bound to say that Lord Lawrence had no

opinion

OPINION (Lat. opinio, from opinari, to think)

of Hodson's integrity in money matters. He has often discussed Hodson's character in talking to me, and it was to him a regret that a man possessing so many fine gifts should have been wanting in a moral quality which made him untrustworthy." Finally, on one occasion Hodson spent
Soo of the pay due to Lieutenant Gcdby, and under threat of exposure was obliged to borrow the money from a native banker through one of his officers named Bisharat Ali.
It was just at the time when Hodson's career seemed ruined that the Indian Mutiny broke out, and he obtained the opportunity of rehabilitating himself. At the very outset of the campaign he made his name by riding with despatches from General Anson at Karnal to Meerut and back again, a distance of 152 M. in all, in seventy-two hours, through a country swarming with the rebel cavalry. This feat so pleased the commanderin-

chief

CHIEF (from Fr. chef, head, Lat. caput)

that he empowered him to raise a regiment of 2000 irregular horse, which became known to fame as Hodson's Horse, and placed him at the head of the Intelligence Department. In his

double

role of cavalry leader and intelligence officer, Hodson played a large part in the reduction of Delhi and consequently in saving India for the British empire. He was the finest swordsman in the army, and possessed that daring recklessness which is the most useful quality of leader-ship against Asiatics. In explanation of the fact that he never received the Victoria Cross it was said of him that it was because he earned it every day of his life. But he also had the defects of his qualities, and could display on occasion a certain cruelty and callousness of disposition. Reference has already been made to Bisharat Ali, who had lent Hodson money. During the siege of Delhi another native, said to be an enemy of Bisharat Ali's, informed Hodson that he had turned rebel
therefore
dE _dn di-
dzxd2y _d2z (1).
ate ate ate
Also, ifs be the arc of the hodograph,
dograph,
dl -V - V ~dt d > 2+ (di/
2'
= V (lf)2+(dt'y)2 L r(d7Z)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.

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