The simultaneous system of two quadratic forms ai, ay, say f and 0, consists of six forms, viz.
If the two things are different, they are either simultaneous or in succession.
Heating or exposure to sunlight reduces it to the red oxide; it fires when ground with sulphur, and oxidizes ammonia to nitric acid, with the simultaneous formation of ammonium nitrate.
Ptolemy knew but of a few latitudes which had been determined by actual observation, while of three longitudes resulting from simultaneous observation of eclipses he unfortunately accepted the least satisfactory, namely, that which placed Arbela 45° to the east of Carthage, while the actual meridian distance only amounts to 34°.
It may denote a simultaneous orthogonal invariant of forms of orders n i, n2, n3,...; degree 0 of the covariant in the coefficients.
The actual formation of this salt is not quite clear; but it is certainly conditioned by the simultaneous contact of decaying nitrogenous matter, alkalies, air and moisture.
Duplex telegraphy consists in the simultaneous transmission of two messages, one in each direction, over the same wire.
In Transylvania, however, the common peril evoked by the Turkish incursion and a simultaneous rising of the Vlach peasantry had knit together the jarring interests of Magyars, Saxons and Szeklers, a union which, under the national hero, the voivode Janos Hunyadi, was destined for a while to turn the tide of war.
When, in the spring of 1769, the first serious campaign was opened by a simultaneous attack by three Russian armies on the principalities, the Crimea and the buffer state of Kabardia, the Turks, in spite of ample warning, were unprepared.
If the invariants and covariants of this composite quantic be formed we obtain functions of X such that the coefficients of the various powers of X are simultaneous invariants of f and 4).
Quadruplex telegraphy consists in the simultaneous transmission of two messages from each end of the line.
The only new problem introduced is the simultaneous transmission of two messages in the same direction; this is sometimes ruplex called " diplex transmission."
A simultaneous insurrection at Massa - Carrara was crushed with similar vigour.
The first chemical change suggested is an interaction between carbon dioxide and water, under the influence of light acting through chlorophyll, which leads to the simultaneous formation of formaldehyde and hydrogen peroxide.
The division in all cases takes place by constriction, or by a simultaneous splitting along an equatorial plane.
But, in actual operation, these two processes are simultaneous; every member is developed as an organ for the performance of some special function.
The closed ovary implies a mode of fertilization which is profoundly different, and which was probably correlated with a simultaneous development of insect life.
The next marked advance in the theory of geography may be taken as the nearly simultaneous studies of the physical earth.
Arrangements had been made for the simultaneous proclamation of Absalom in all parts of the land.
A pilgrimage feast must be fixed in date to ensure the simultaneous presence of the pilgrims. There are, besides, seeming references to the feast in the early prophets, as Hosea xii.
The process is therefore adapted to the simultaneous estimation of carbon,hydrogen, the halogens and sulphur.
An elaborate plan of operations, which he described in detail in a letter to his brother after his arrest, had been prepared by Emmet, the leading feature of which was a simultaneous attack on the castle, the Pigeon House and the artillery barracks at Island bridge; while bodies of insurgents from the neighbouring counties were to march on the capital.
Turkey was at this time the only neutral state in Europe; it was of vital im- Treaty of portance that she should not be absorbed into the Napoleonic system, as in that case Russia would have been exposed to a simultaneous attack from France, Austria, Turkey and Persia.
This process will go on until the simultaneous separation of electric charges produces an electrostatic force strong enough to prevent further separation of ions.
Nevertheless, a simultaneous outbreak of a jacquerie in Little-Russia contributed to the extension of the confederation throughout the eastern province of Poland and even in Lithuania.
We cannot combine the equations so as to eliminate the variables unless on the supposition that the equations are simultaneous, i.e.
This expression of R shows that, as will afterwards appear, the resultant is a simultaneous invariant of the two forms.
From these formulae we derive two important relations, dp4 = or the function F, on the right which multiplies r, is said to be a simultaneous invariant or covariant of the system of quantics.
This notion is fundamental in the present theory because we will find that one of the most valuable artifices for finding invariants of a single quantic is first to find simultaneous invariants of several different quantics, and subsequently to make all the quantics identical.
Since (ab) = a l b 2 -a 2 b l, that this may be the case each form must be linear; and if the forms be different (ab) is an invariant (simultaneous) of the two forms, its real expression being aob l -a l b 0.
May be always viewed as a simultaneous invariant of a number of different linear forms a x, x, c x, ....
May be a simultaneous invariant of a number of different forms az', bx 2, cx 3, ..., where n1, n 2, n3, ...
In general it will be simultaneous covariant of the different forms n 1 rz 2 n3 a, b x, ?
For the cubic (ab) 2 axbx is a covariant because each symbol a, b occurs three times; we can first of all find its real expression as a simultaneous covariant of two cubics, and then, by supposing the two cubics to merge into identity, find the expression of the quadratic covariant, of the single cubic, commonly known as the Hessian.
- We have seen that (ab) is a simultaneous invariant of the two different linear forms a x, bx, and we observe that (ab) is equivalent to where f =a x, 4)=b.
(a m b n) k (ab) kamkbn-k x, x - x it is clear that the k th transvectant is a simultaneous covariant of the two forms.
It will be shown later that all invariants, single or simultaneous, are expressible in terms of symbolic products.
As regards simultaneous binary forms, the system of two quadratics, and of any number of quadratics, is alluded to above and has long been known.
2 cos w xy+y 2 = X 2 +2 cos w'XY+Y2, from which it appears that the Boolian invariants of axe+2bxy-y2 are nothing more than the full invariants of the simultaneous quadratics ax2+2bxy+y2, x 2 +2 cos coxy+y2, the word invariant including here covariant.
In general the Boolian system, of the general n i °, is coincident with the simultaneous system of the n i °' and the quadratic x 2 +2 cos w xy+y2.
These are knocked off, ground up with water, freed from metal-particles by elutriation, and the paste of white lead is allowed to set and dry in small conical forms. The German method differs from the Dutch inasmuch as the lead is suspended in a large chamber heated by ordinary means, and there exposed to the simultaneous action of vapour of aqueous acetic acid and of carbon dioxide.
Various currents are then passed through the magnetizing coil, the galvanometer readings and the simultaneous magnetometer deflections being noted.
Honda subjected tubes of iron, steel and nickel to the simultaneous action of circular and longitudinal fields, and observed the changes of length when one of the fields was varied while the other remained constant at different successive values from zero upwards.
Invasion and conquest was the simultaneous Catholic reaction in Hungary.
Simultaneous equations in two unknowns x and y may be treated in the same way, except that each equation gives a functional relation between x and y.
His fleet, too, lingered uselessly in the harbour, till after a defeat in which Eurymedon perished, though the simultaneous land attack was unsuccessful.
The result of his discovery was an entire revolution in the knowledge of diseases of the chest; but it would be a mistake to forget that an essential factor in this revolution was the simultaneous study of the condition of the diseased organs as seen after death.
Calcium cyanamide has assumed importance in agriculture since the discovery of its economic production in the electric furnace, wherein calcium carbide takes up nitrogen from the atmosphere to form the cyanamide with the simultaneous liberation of carbon.
Malatesta contributed to the defeat by preventing a simultaneous attack by the besieged.
The south-westerly winds which prevail north of the equator during the hot half of the year, to which navigators have given the name of the south-west monsoon (the latter word being a corruption of the Indian name for season), arise from the great diminution of atmospheric pressure over Asia, which begins to be strongly marked with the great rise of temperature in April and May, and the simultaneous relatively higher pressure over the equator and the regions south of it.
The investigation of the properties of these functions, as well for a single form as for a simultaneous set of forms, and as well for one as for many series of variables, is included in the theory of invariants.
If ai, bx, cx be different forms we obtain, after development of the squared determinant and conversion to the real form (employing single and double dashes to distinguish the real coefficients of bx and cz), a(b'c"+b"c'-2 f'f") +b(c'a"+c"a'-2g'g") +c(a' +a"b'-2h'h")+2f(g'h"+g"h'-a' + 2g (h ' f"+h"f'-b'g"-b"g')+2h(f'g"+f"g'-c'h"-c"h'); a simultaneous invariant of the three forms, and now suppressing the dashes we obtain 6 (abc+2fgh -af t - bg 2 -ch2), the expression in brackets being the S well-known invariant of az, the vanishing of which expresses the condition that the form may break up into two linear factors, or, geometrically, that the conic may represent two right lines.
He successfully considers the systems of two and three simultaneous ternary quadratics.
Having come to an understanding with his father-in-law Podébrad, he was able to turn his arms against the emperor Frederick, and in April 1462 Frederick restored the holy crown for 60,000 ducats and was allowed to retain certain Hungarian counties with the title of king; in return for which concessions, extorted from Matthias by the necessity of coping with a simultaneous rebellion of the Magyar noble in league with Podebrad's son Victorinus, the emperor recognized Matthias as the actual sovereign of Hungary.
It is readily decomposed by water with formation of sulphurous, sulphuric and thiosulphuric acids, with simultaneous liberation of sulphur.
Solutions of persulphates in the cold give no precipitate with barium chloride, but when warmed barium sulphate is precipitated with simultaneous liberation of chlorine: K 2 S 2 0 8 + BaC1 2 = BaSO 4 + K 2 SO 4 + C1 2.
If simultaneous, cause is effect and effect cause.
This simultaneous discussion of many topics did not prevent a clear understanding but on the contrary was the surest sign that they fully understood one another.
When, for instance, we say that Napoleon ordered armies to go to war, we combine in one simultaneous expression a whole series of consecutive commands dependent one on another.