Yet so long as the caliphs lived in Medina, the capital of Arabia was the capital of the expanding Arabian empire.
X (1 +PD1+12D2+...+ï¿½8D8+...) fm, and now expanding and equating coefficients of like powers of /t D 1 f - Z(Difi)f2f3.
It resembles in colour some varieties of the latter snake, and, like this, it has the power, though in a less degree, of expanding its hood.
In the work of expanding and elaborating this theme the successors of Geoffrey went as far beyond him as he had gone beyond Nennius; but he retains the credit due to the founder of a great school.
The remainder is equivalent to the external work, W, done by the body in expanding or otherwise, which can be utilized for mechanical purposes, and ceases to exist as heat in the body.
The area BCDdb under the path represents the external work done by the substance in expanding from B to D, which is analytically represented by the integral of pdv taken along the given path.
Observing that F is a function of the co-ordinates expressing the state of the substance, we obtain for the variation of S with pressure at constant temperature, dS/dp (0 const) '=' 2 F/dedp =-0d 2 v/d0 2 (p const) (12) If the heat supplied to a substance which is expanding reversibly and doing external work, pdv, is equal to the external work done, the intrinsic energy, E, remains constant.
Joule failed to observe any change of temperature in his apparatus, and was therefore justified in assuming that the increase of intrinsic energy of a gas in isothermal expansion was very small, and that the absorption of heat observed in a similar experiment in which the gas was allowed to do external work by expanding against the atmospheric pressure was equivalent to the external work done.
For several centuries Tegea served as a bulwark of Arcadia against the expanding power of Sparta; though ultimately subdued about 550 B.C. it was allowed to retain its independence and its Arcadian nationality.
It consists of a small hydraulic press, which forces a set of expanding bits or wedges into a bore-hole previously bored by a long screw augur or drill, worked by hand, the action of the press being continued until a sufficient strain is obtained to bring down the coal.
Or, expanding as far as first powers of dt, dpidqi.
The equation of energy is dQ=dE+pdv, (17) expressing that the total energy dQ is used partly in increasing the internal energy of the gas, and partly in expanding the gas against the pressure p. If we take p = RNT/v from equation (14) and substitute for E from equation (16), this last equation becomes dQ 2 (n +3)RNdT +RNTdv (18) which may be taken as the general equation of calorimetry, for a gas which accurately obeys equation (14).
Doubtless with the object of expanding the flourishing foreign trade of Samos, he entered into alliance with Amasis, king of Egypt, who, according to Herodotus, renounced his ally because he feared that the gods, in envy of Polycrates' excessive good fortune, would bring ruin upon him and his allies.
Subsequent photographs showed that this nebula, which consisted mainly of two incomplete rings of nebulosity, was expanding outwards at the rate of from 2" to 3" per day.
Quinn and Martha perpetually had their hands full with their baby and Betsy stayed home, content to have extra time with our expanding garden.
He rested his hand on her expanding stomach and his chin on her head.
He even produced a notebook, a sure sign of expanding curiosity.
He flicked buttons, zooming and expanding the scenes and adjusting it until it was as he preferred it: spinning faster than normal with alternating close-ups of each major battle.
Katie said, rising. The ground still rumbled, the trees surrounding both food cubes expanding fast and tearing up the ground in several directions as they did. She looked around, irritated to find she'd caused a chasm to form between them and the direction they'd been running.
She ran a hand across her expanding stomach.
In 1839 Seguin endeavoured to determine the mechanical equivalent of heat from the loss of heat suffered by steam in expanding, assuming that the whole of the heat so lost was consumed in doing external work against the pressure to which the steam was exposed.
The amount of heat absorbed by the air could thus be measured, while the work done by it in expanding could be readily calculated.
He shows that the amount of work obtainable is equal to that which can be done by the first gas in expanding into the space occupied by the second (supposed vacuous) together with that done by the second in expanding into the space occupied by the first.
These static and kinetic conditions succeed each other rapidly, and the result is to detach or throw off from the antenna semi-loops of electric force, which move outwards in all directions and are accompanied by expanding circular lines of magnetic force.
There was little of originality in Luria's doctrines; the theory of emanations, the double belief in the process of the Divine Essence as it were self-concentrating (Zimzum) and on the other hand as expanding throughout creation; the philosophical " sceptism '° which regards God as unknowable but capable of direct intuition by feeling - these were all common elements of mystical thought.
The great summer heat, by expanding the air upwards, disturbs the level of the planes of equal pressure, and causes an outflow of the upper strata from the heated area.
The present magnificent building was a slow growth extending over three centuries and expanding gradually as the republic grew in riches.
The history of Venice during the next two hundred years is marked externally by the growth of the city, thanks to an ever-expanding trade, both down the Adriatic, which brought the republic into collision with the Dalmatian pirates and led to their final conquest, in 1000, by the doge Pietro Orseolo II., and also on the mainland, where Venice gradually acquired trading rights, partly by imperial diploma, partly by the establishment and the supply of markets on the mainland rivers, the Sile and the Brenta.
Contemporaneously other events were menacing the ascendancy and exhausting the treasury of the republic. In 1453 Constantinople fell to the Ottoman Turks, and although Venice entered at once into treaty with the new power and desired to trade with it, not to fight with it, yet it was impossible that her possessions in the Levant and the archipelago should not eventually bring her into collision with the expanding energy of the Mussulman.
This consists in the use of an expanding reamer by means of which Drilling in the well may be drilled to a diameter admitting of the Galicia.
But the extension of properties in the hands of the patricians, and the continued absences of citizens required by the expanding system of conquest, necessarily brought with them a demand for slave labour, which was increasingly supplied by captives taken in war.
+amam Expanding the right-hand side by the exponential theorem, and then expressing the symmetric functions of al, a2, ...a m, which arise, in terms of b1, b2, ...'
Now log (1+ï¿½X1 +/22X2+/ï¿½3X3 +ï¿½ï¿½ï¿½) =E log (1+/2aix1+22aix2-1-/23ax3+...) whence, expanding by the exponential and multinomial theorems, a comparison of the coefficients of ï¿½n gives (n) (-)v1+v2+v3+..
The existence of such a relation, as 0-1+0-2+.,.+cr2=0, necessitates the vanishing of a certain function of the coefficients A2, A 3, ...A 9, and as a consequence one product of these coefficients can be eliminated from the expanding form and no seminvariant, which appears as a coefficient to such a product (which may be the whole or only a part of the complete product, with which the seminvariant is associated), will be capable of reduction.
Descending series of the semi-convergent class, available for numerical calculation when u is moderately large, can be obtained from (12) by writing x=uy, and expanding the denominator in powers of y.
This means very much, though his modesty led him to call in the aid of his friend Saul to cope with the new and expanding situation (25 f.).
When a quantity of heat, H, is supplied to a body, part is expended in raising the temperature of the body, or in expanding the volume against molecular forces, and is represented by an increase in the total quantity of energy contained in the body, which is generally called its Intrinsic Energy, and will be denoted by the symbol E.
Business interests of this magnitude were constantly expanding, and he became interested in numerous subsidiary enterprises, such as tramways and the supply of electric power and light.
It is at the head of a narrow defile, shut in by steep hills on the east and west but expanding on the north to meet the valley of the Save.
To the south the London basin is succeeded by the North Downs, an elevated ridge of country consisting of an outcrop of chalk which extends from Westerham to Folkestone with an irregular breadth generally of 3 to 6 miles, but expanding to nearly 12 miles at Dartford and Gravesend and also to the north of Folkestone.
The latter integral becomes, on expanding in a series, fds/V f(udx+vdy+wdz)/V2-1-f(udx+vdy+wdz)2/V3ds+..., since ids = dx.
This bank continues southwards to the Antarctic Ocean, expanding into a plateau on which Australia stands, and a branch runs eastwards and then southwards from the north-east of Australia through New Zealand.
Thermometer, since, fluids generally expanding more than the solids of which the instrument is constructed, the instrument will sink as the temperature rises.
It passes by the hamlet of Port Valais, once on the shore of the lake, before expanding into the Lake of Geneva, between Villeneuve (right) and St Gingolph (left).
The Chronicle still survived as a medium of conveying information, though more often than not this was now written by a layman; but new stores of information were coming into existence, or rather the old stores were expanding and taking a different form.
It is simply a hydrostatic mechanism for expanding the tentacles.
In this capacity he was also ex officio the superintendent of common schools, and rendered valuable services to his state in perfecting and expanding the free public school system, and in establishing state normal schools.
The secular importance of Henry's activity has been somewhat obscured by his achievements in the sphere of ecclesiastical politics; but no small part of his energies was devoted to the task of expanding the royal authority at the expense of temporal competitors.
He believed that life was an expanding, growing force, and that animals responded to the environment by developing new wants, seeking to satisfy these by new movements and thus by their own striving producing new organs which were transmitted to their descendants.
In his speech to the delegations in 1898 he dwelt on the necessity of expanding Austria's mercantile marine, and of raising the fleet to a strength which, while not vying with the fleets of the great naval powers, would ensure respect for the Austrian flag wherever her interests needed protection.
He was the last of those universal minds which have been able to compass all domains of human activity and knowledge; for he stood on the brink of an era of rapidly expanding knowledge which has made for ever impossible the universality of interest and sympathy which distinguished him.
These fusible metals have the peculiarity of expanding as they cool; Rose's metal, for instance, remains pasty for a considerable range of temperature below its fusing point, contracts somewhat rapidly from 80° to 55°, expands from 55° to 35°, and contracts again from 35° to o°.
It has been said that the declared object of this policy had been to maintain the independence and integrity of Afghanistan, to secure the friendly alliance of its ruler, and thus to interpose a great barrier of mountainous country between the expanding power of Russia in Central Asia and the British dominion in India.
It then takes in heat from the furnace, expanding in volume and forcing the piston (B) to rise, which completes the cycle.
Pumpkin Green was seen about town, always in the company of ever-expanding Melissa, whom he seemed to adore.
One hand rested on her expanding stomach.
Has he been expanding his clientele?
Came less, while at the same time the expanding population necessarily led to diversified interests at home.