In order to separate the distillate into various fractions, and to remove as much of it as possible free from condensed steam, it is now usual to employ condensing appliances of special form with outlets for running off the different fractions.
Obviously, therefore, liquids are comparable when the pressures, volumes and temperatures are equal fractions of the critical constants.
The Liber abaci, which fills 459 printed pages, contains the most perfect methods of calculating with whole numbers and with fractions, practice, extraction of the square and cube roots, proportion, chain rule, finding of proportional parts, averages, progressions, even compound interest, just as in the completest mercantile arithmetics of our days.
2 We have given the fractions in the order in which they occur in the modern system.
On the side of the defence, each colonel had been left to retire as best he could, and thus certain fractions of the retreating Russians encountered Inouye's advancing troops and were destroyed after a most gallant resistance.
(c) the standard gallon (and multiples and fractions of it), declared to contain 10 lb of water at 62° F., being in volume 277.274 cub.
Napier thus had complete command over decimal fractions and the use of the decimal point.
Benzene is manufactured from the low-boiling fractions of the coal-tar distillate.
The accuracy of a meter is tested by drawing calibration curves showing the percentage departure from absolute accuracy in its reading for various decimal fractions of full load.
Was to divide his territories among his sons, whereby Poland was partitioned into no fewer than four, and ultimately into as many as eight, principalities, many of which (Silesia and Great Poland, for instance) in process of time split up into still smaller fractions all of them more or less bitterly hostile to each other.
He extended the "law of continuity" as stated by Johannes Kepler; regarded the denominators of fractions as powers with negative exponents; and deduced from the quadrature of the parabola y=xm, where m is a positive integer, the area of the curves when m is negative or fractional.
2), with a float plate on its upper side, carrying three indicating dials, recording respectively fractions, units and tens of miles (up to a hundred).
For forty years after the death of its founder it remained united under the authority of a series of grand khans chosen from among his descendants, and then it began to fall to pieces till the various fractions of it became independent khanates.
Beneath are the official Liverpool quotations of " futures," as they appeared on the morning of the 19th of April 1906: A merican Deliveries, any port, basis of middling, good ordinary clause (the fractions are given in moths of a penny).
If we express the pressure, volume and temperature as fractions of the critical constants, then, calling these fractions the " reduced " pressure, volume and temperature, and denoting them by 7r, 0 and 0 respectively, the characteristic equation becomes (7+3/0 2)(30-i) =80; which has the same form for all substances.
For the subjects of this heading see the articles DIFFERENTIAL EQUATIONS; FOURIER'S SERIES; CONTINUED FRACTIONS; FUNCTION; FUNCTION OF REAL VARIABLES; FUNCTION COMPLEX; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; MAXIMA AND MINIMA; SERIES; SPHERICAL HARMONICS; TRIGONOMETRY; VARIATIONS, CALCULUS OF.
The amounts of Turkish gold, silver and debased coinage in circulation are approximately £T16,500,000, in gold, £T8,70o,000 (940,000,000 piastres at 108) in silver mejidies and fractions, and 200,000,000 piastres in beshlik and metallik.
We have to multiply a01; -alas+a2 by ao, -aif32+a2 and we obtain ao (3 - aoal(f31N2 +01133) +aoa2(SI+13) -i-a?31a2 - aIa2(31 + 02) + al, 131+02 = b, 131132 = b t'i +s2 = 2bob2, and clearing of fractions R 1,5 = (a o b 2 - a2 b o) 2 + (a i b o - aobi) (aib2 - a2b1).
In astronomical practice the masses of the planets are commonly expressed as fractions of the mass of the sun, the latter being taken as unity.
Unfortunately these eclipses are not sudden but slowly changing phenomena, so that they cannot be observed without an error of at least several seconds, and not infrequently important fractions of a minute.
In algebra he discovered the method of approximating to the real roots of an equation by means of continued fractions, and imagined a general process of solving algebraical equations of every degree.
(v.) The use of the solidus / separating two numbers is for convenience of printing fractions or fractional numbers.
The deductions follow directly from the definitions, and such mechanical processes as "clearing of fractions " find no place (ï¿½ 21 (ii.)).
(ii.) Perhaps the worst thing we can do, from the point of view of intelligibility, is to " clear of fractions " by multiplying both sides by 6.
(iii.) The general statement of the laws of operation of fractions is perhaps best deferred until we come to fractional numbers, when letters can be used to express the laws of multiplication and division of such numbers.
(iii.) There are important theorems as to the relative value of fractions; e.g.
The idea of (-5) as a number with which we can perform such operations as multiplication comes later (ï¿½ 49)ï¿½ (ii.) On the other hand, the conception of a fractional number follows directly from the use of fractions, involving the subdivision of a unit.
We find that fractions follow certain laws corresponding exactly with those of integral multipliers, and we are therefore able to deal with the fractional numbers as if they were integers.
Thus, if we have an equation P=Q, where P and Q are numbers involving fractions, we can clear of fractions, not by multiplying P and Q by a number m, but by applying the equal multiples P and Q to a number m as unit.
ï¿½; the successive terms of this series, after the first, are alternately positive and negative, and consist of fractions with numerators I and denominators continually increasing.
So persistently does the human ear rebel against the division of the tetrachord into two greater tones and a leimma or hemitone, as represented by the fractions 9, 9, 26, that, centuries before the possibility of reconciling the demands of the ear with those of exact science was satisfactorily demonstrated, the Aristoxenian school advocated the use of an empirical scale, sounding pleasant to the sense, in preference to an unpleasing tonality founded upon immutable proportions.
The fractions are agitated with strong sulphuric acid, and then washed with a caustic soda solution.
Generally the components of a mixture will be vaporized in the order of their boiling-points; consequently if the condensates or "fractions" corresponding to definite ranges of temperature be separately collected, it is obvious that a more or less partial separation of the components will be effected.
For the purpose of collecting the distillates in fractions, many forms of receivers have been devised.
Dephlegmation of the vapours arising from such mixtures as coaltar fractions, petroleum and the "wash" of the spirit industry, is very important, and many types of apparatus are employed in order to effect a separation of the vapours.
Both the fractions into which they were divided by the Nerbudda river laid claim to antiquity: while the northern, however, did not trace their origin further back than the period of the early Mahommedan kings of Delhi, the southern fraction not only claimed an earlier and purer descent, but adhered also with greater strictness to the rules of their profession.
This was arranged for by a movable leaf carrying the sighting V, worked by means of a mill-headed screw provided with a scale in degrees and fractions to the same radius as the elevation scale, and an arrowb head for reading.
The Army of the North was to concentrate in three fractions - around Solre, Beaumont and Philippeville - as close to Charleroi as was practicable; and he arranged to screen the initial movements of the troops as much as possible, so as to prevent the allies from discovering in time that their centre was aimed at.
The time between the breaks could be measured in seconds by the clock signals, and in fractions of a second by the tuning-fork record.
The frequency ratios in the diatonic scale are all expressible either as fractions, with i, 2, 3 or 5 as numerator and denominator, or as products of such fractions; and it may be shown that for a given note the numerator and denominator are smaller than any other numbers which would give us a note in the immediate neighbourhood.
Thus, whilst the detachment was still disposed as a series of rearguards, the foremost fractions of it stood to fight on the Yalu, against odds of four to one.
Calling in the brigade detached to the assistance of Nozu as well as all other available fractions of his scattered army, he himself attacked 1 The 5th division of the 2nd Army had been sent to join the 10th as the latter approached Hsimucheng.
Logistic or Proportional Logarithms. - The old name for what are now called ratios or fractions are logistic numbers, so that a table of log (a/x) where x is the argument and a a constant is called a table of logistic or proportional logarithms; and since log (a/x) =log a-log x it is clear that the tabular results differ from those given in an ordinary table of logarithms only by the subtraction of a constant and a change of sign.
Be proper fractions, and the value of every one of the interminate continued fractions l!1 a2.
Which Gives The Series Of Approximating Fractions, I 7 8 31 132 163 &C.
Decimal fractions had been employed for the extraction of square roots.
The point separating the integers from the decimal fractions seems to be the invention of Bartholomaeus Pitiscus, in whose trigonometrical tables (1612) it occurs and it was accepted by John Napier in his logarithmic papers (1614 and 1619).
The size of the angle between the median planes of two consecutive leaves in an alternate arrangement is their divergence; and it is expressed in fractions of the circumference of the axis which is supposed to be a circle.
AS, the Roman unit of weight and measure, divided into 12 unciae (whence both "ounce" and "inch"); its fractions being deunx i 2, dextans, dodrans 4 i bes 3, septunx T7-2-, semis z, quincunx A, triens 3 i quadrans 4, sextans s, sescuncia $, uncia r i g.
Laurent generally agreed, except when the theory compelled the adoption of formulae containing fractions of atoms; in such cases he regarded the molecular weight as the weight occupying a volume equal to four unit weights of hydrogen.
Leonardo, making use of fractions of the sexagesimal scale, gives X = I° 221 7 42" i 33 iv 4v 40 vi, after having demonstrated, by a discussion founded on the 10th book of Euclid, that a solution by square roots is impossible.
There are extensions of the binomial theorem, by means of which approximate calculations can be made of fractions, surds, and powers of fractions and of surds; the main difference being that the number of terms which can be taken into account is unlimited, so that, although we may approach nearer and nearer to the true value, we never attain it exactly.
2 an embolismic The result so obtained would in general be more accurate than the Jewish calculation, from which it may differ a day, as fractions of a day do not enter alike in these computations.
African mines the diamonds are not only crystals of various weights from fractions of a carat to 150 carats, but also occur as microscopic crystals disseminated through the blue ground.
Thus, to take the latter one, if we suppose that of two editors of equal competence A requires a probability of four-fifths to admit a reading into his text and B a probability of three-fifths only, then in all the cases in which the probability lies between these two fractions B will be right seven times to A's three, while outside these limits there will be no difference between them.
If each of the fractions (3) is put equal to i/4h, it is readily found, from the first property of the normal state, that, of the s molecules of the first kind, a number sal (h3m3 /13)e hm (u2+v2+w2)dudvdw (4) Velocities.
Recorde published several works upon mathematical subjects, chiefly in the form of dialogue between master and scholar, viz.: - The Grounde of Artes, teachinge the Worke and Practise of Arithmeticke, both in whole numbers and fractions (1540); The Pathway to Knowledge, containing the First Principles of Geometry.