745 



STADE DUTIES. 



STADIUM. 



746 



tion, and the equilibrium is unstable ; in the third, displacement 

 would only be a removal to another position of rest, and the equili- 

 brium is called indifferent. In the fourth, displacement to the right 

 would be followed by restoration, but the velocity acquired in restora- 

 tion would carry the molecule to the left, on which side there is no 

 tendency to restoration : the equilibrium would then be permanently 

 disturbed, and practically unstable ; though it might be convenient to 

 say that it IB stable as to the disturbances to the right, and unstable as 

 to those to the left. In the fifth, the equilibrium at A is unstable, 

 but if a push, however slight, were given to the molecule, it would 

 obviously, by reason of the two contiguous stable positions, oscillate 

 about A, as" if A were itself a stable position : and in the same 

 manner a stable position, with an unstable one near to it, might, for a 

 disturbance of sufficient magnitude, present the phenomena of an un- 

 stable position. 



Now, suppose that the point A, instead of being a single molecule, 

 is the centre of gravity of a system acted on by its own weight only ; 

 and let the curve drawn be the path of the centre of gravity, which, 

 owing to the connection of the parts of the system with its supports, 

 that centre is obliged to take. The phenomena of the single point 

 still remain true : there is in every case a position of equilibrium 

 when the system is placed in such a position that its centre of gravity 

 is at A. In (1) the equilibrum is stable; in (2), unstable; in (3), 

 indifferent; in (4), stable or unstable, according to the direction of 

 disturbance ; in (5), unstable, with results like those of stability. It 

 is an error to state, as is frequently done, that there is no equilibrium 

 in such a system except when its centre of gravity is highest or 

 lowest ; as is obvious from (3) and (4). The general proposition which 

 t true is this that a system acted on by its own weight is in equili- 

 brium then, and then only, when its centre of gravity is placed at that 

 point of its path which has its tangent parallel to the horizon, or per- 

 pendicular to the direction of gravity. 



When a system U supported on three or more points, it is well 

 known that there is no equilibrium unless the vertical passing 

 through the centre of gravity cuts the polygon formed by joining 

 these points. This must not be confounded, as is sometimes done, 

 with a case of distinction between stable and unstable equilibrium ; 

 for it is a case of equilibrium or no equilibrium, according as the 

 central vertical cuts or does not cut the base of the figure. Of course 

 it is in the power of any one to say that stability means equilibration 

 and instability non-equilibration ; but such is not the technical use of 

 these words in mechanics : stability : and instability refer to equili- 

 brium, stable equilibrium being that which would only be converted 

 into oscillation by a disturbance, and unstable equilibrium that which 

 would not be so converted. 



Neither must the effects of friction or other resistances be con- 

 founded with those of a stable or unstable disposition. A ladder rest- 

 ing against horizontal ground and a vertical wall is maintained by 

 friction ; were it not for friction, there would not be rest hi any posi- 

 tion ; and as it is, the angle which the ladder makes with the ground 

 must not be too small. There is thus a set of positions, from the 

 vertical one to a certain inclination, depending on the amount of 

 friction, in all of which there is equilibrium ; while in every other 

 position there is no equilibrium. In no case must the words stability 

 and instability be used in such manner as to confuse their popular 

 with their technical sense. 



Under SOLAR S YSTKM is pointed ont what is meant by the stability 

 of that system. When a system has a motion of a permanent charac- 

 ter, it is stable if a small disturbance only produce oscillations in that 

 motion, or make permanent alterations of too slight a character to 

 allow the subsequent mutual actions of the parts to destroy the per- 

 manent character of the motion. Suppose a material body, for 

 instance, to revolve about an axis passing through the centre of gravity 

 unacted on by any forces except the weight of its parts. If this 

 axis be one of the principal axes, the rotation on it is permanent, that 

 i.", the axis of rotation will continue unaltered, even though that axis 

 be not fixed. The rotation however, though permanent, is not stable 

 about more than two out of the three principal axes. Let the first 

 rotation be established about the axis which has the greatest moment 

 of rotation, or the least, and if a slight displacement or disturbance be 

 given, which has the effect of producing a little alteration of the axis 

 of rotation, that alteration will not increase indefinitely, but will only 

 occasion a perpetual transmission of the rotation from axis to axis, all 

 the lines lying near to the principal axis first mentioned. But if that 

 axis be chosen about which the moment of inertia is neither greatest 

 nor least, any disturbance, however slight, will continually remove the 

 axis of rotation farther and farther from the first axis, near which it 

 will not return until it has made a circuit about one of the other two 

 principal axes. 



For the mathematical part of this subject, so far as we give it, see 

 VinTCAt VELOCITIES. 



STADE DUTIES are so called from Stade, in the kingdom of 

 Hanover, a town situated on the right bank of the Schwinge, three or 

 four miles from where it falls into the Elbe, and 22 miles west by 

 north from the city of Hamburg. The name Srunshauseu Tolh is now 

 more commonly used, from the village of Brunshausen, at the mouth 

 of the Schwinge, where there is a custom-house and a royal guard-ship, 

 and when: the duties are collected which are levied on vessels and 



merchandise passing up the Elbe. The original duties, which were 

 regulated by a treaty made in 1691, were light, but were gradually 

 increased by the Hanoverian government till they amounted to about 

 40,000/. a-year. The duties levied were about J per cent, ad valorem, 

 more on some articles and less on others. British vessels by a procla- 

 mation of Geo. II., December 1, 1736, were allowed under certain 

 regulations to sail directly up to Hamburg, without coming to anchor 

 at the mouth of the Schwinge, as other foreign vessels were obliged 

 to do. 



By a convention between the King of Hanover and the heads of the 

 other Elbe-bordering states (Emperor cf Austria, King of Prussia, 

 King of Saxony, King of Denmark, Duke of Mecklenburg-Schwerin, 

 Duke of Anhalt-Coethen, Duke of Anhalt-Dessau, Duke of Anhalt- 

 Bernburg, Free and Hauseatic town of Lubeck, and Free and Hanseatic 

 town of Hamburg), dated April 13, 1844, in conformity with articles 

 108 and 116 of Act of Congress of Vienna, of June 9, 1815, the 

 Brunshausen Tolls were revised, regulated, and settled according to a 

 Toll-Tariff agreed upon by the contracting parties. 



These rates, by a treaty with Great Britain in July, 1844, were to 

 continue in force till January 1, 1854. The treaty was then renewed, 

 temporarily, while the question of the abolition of the duties was 

 considered, as they were felt to be oppressive and unjust, nothing 

 being done to benefit the navigation in return for them. At length a 

 compensation of 3,000,00(M. was offered, of which Hamburg, which 

 would derive the greatest benefit, was to pay one-third ; Great Britain 

 l,125,20ti/., and all the other commercial states of Europe with the 

 United States of America the remainder, in proportion to their com- 

 merce. This has been agreed to, but the ratification has not yet been 

 completed. 



STA'DIUM (& ffraStos and rb tmiSiov), the principal Greek measure of 

 length, was equal to 600 Greek or 625 Roman feet, that is, to 606 feet 

 9 inches English. The Roman mile contained 8 stadia. The Roman 

 writers often measure by stadia, chiefly in geographical and astro- 

 nomical measurements. (Herod., ii. 149 ; Plin., ' Hist. Nat./ ii., 23 or 

 21 ; Columell., ' Re. Rust.,' v. 1 ; Strabo, vii., p. 497.) 



The standard length of this measure was the distance between the 

 pillars at the two ends of the foot-race course at Olympia, which was 

 itself called stadium, from its length, and this standard prevailed 

 throughout Greece. Some writers have attempted to show that there 

 were other stadia in use in Greece besides the Olympic. The only 

 passages in which' anything of the kind seems to be stated are one 

 in Censorinus ('De Die Natali,' c. 13), which, as far as it can be 

 understood, evidently contains some mistake; and another which 

 is quoted by Aulus Gellius (i. 1) from Plutarch, but which 

 speaks of the race-courses called stadia, not of the stadium as a 

 measure. 



The principal argument for a variety of stadia is that of Major 

 Rennell (' Geog. of Herod ,' s. 2) ; namely, that when ancient authors 

 have stated the distances between known places, and a comparison is 

 made between their statements and the actual distances, the distances 

 stated by them are invariably found to be too great, never too small. 

 Hence the conclusion is drawn that they used an itinerary stade shorter 

 than the Olympic. If so, it is strange that the very writers who have 

 left us these statements of distances have not said a word about 

 the itinerary stade which they are supposed to have used, while several 

 of them often speak of the Olympic stade as containing 600 Greek feet. 

 But there is a very simple explanation of the difficulty, which is given 

 by Ukert, in his ' Geographic der Griechen und Rbmer ' (i., ii., p. 56, 

 Ac.). The common Greek method of reckoning distances, both by sea 

 and land, was by computation, not by measurement. A journey or 

 voyage took a certain number of days, and this number was reduced 

 to stadia, by allowing a certain number of stadia to each day's journey. 

 The number of stadia so allowed was computed on the supposition that 

 circumstances were favourable to the traveller's progress ; and there- 

 fore every impediment, such as wind, tide, currents, windings of the 

 coast, a heavily laden or badly sailing ship, or any deviation from the 

 shortest track by sea, and the corresponding hindrances by land, would 

 all tend to increase the number of days which the journey took, and 

 consequently the number of stadia which the distance was computed 

 to contain. These circumstances, together with the fact that the Greek 

 writers are by no means agreed as to the number of stadia contained 

 in a day's journey, and other sources of inaccuracy which we know to 

 have existed, furnish a satisfactory explanation of the discrepancies 

 which we find in their statements of distances, both when compared 

 with one another, and when compared with the actual fact, without 

 there being any occasion to resort to the supposition of a stade different 

 from the Olympic. Colonel Leake ' On the Stade as a Linear Measure ' 

 ('Journal of the Royal Geographical Society of London,' vol. ix., 

 1839), has also come to the conclusion "that the stade, as a linear 

 measure, had but one standard, namely, the length of the foot-race, 

 or interval between the iupfT-ijpia and o/pr7)/> in all the stadia of 

 Greece, and which is very clearly defined as having contained 600 

 Greek feet." 



When we come however to writers as late as the 3rd century of the 

 Christian era, we do find stadia of different lengths. Of these the 

 chief are those of 7 and 7i to the Roman mile. (Wurm, ' De Pond. ' 

 &c., 58.) 



The following table, from the Appendix to Hussey's 'Ancient 



