But if the given curve has a node, then not only the Hessian passes through the node, but it has there a node the two branches at which touch respectively the two branches of the curve; and the node thus counts as six intersections; so if the curve has a cusp, then the Hessian not only passes through the cusp, but it has there a cusp through which it again passes, that is, there is a cuspidal branch touching the cuspidal branch of the curve, and besides a simple branch passing through the cusp, and hence the cusp counts as eight intersections.
This method of solution fails when the discriminant R vanishes, for then the Hessian has equal roots, as also the cubic f.
The Hessian in that case is a factor of f, and Q is the third power of u2,...
The quartic has four equal roots, that is to say, is a perfect fourth power, when the Hessian vanishes identically; and conversely.
This can be verified by equating to zero the five coefficients of the Hessian (ab) 2 axb2.
Gordan has also shown that the vanishing of the Hessian of the binary n ic is the necessary and sufficient condition to ensure the form being a perfect n th power.
The quartic will have two pairs of equal roots, that is, will be a perfect square, if it and its Hessian merely differ by a numerical factor.
For it is easy to establish] the formula (yx) 2 0 4 = 2f.4-2(f y 1) 2 connecting the Hessian with the quartic and its first and second polars; now a, a root of f, is also a root of Ox, and con se uentl the first polar 1 of of q y p f?
-, reduce s x2ax1 -x10x2 to the form j Oz ON 2 1 1 j 2 i The Binary Quintic.-The complete system consists of 23 forms, of which the simplest are f =a:; the Hessian H = (f, f') 2 = (ab) 2axbz; the quadratic covariant i= (f, f) 4 = (ab) 4axbx; and the nonic co variant T = (f, (f', f") 2) 1 = (f, H) 1 = (aH) azHi = (ab) 2 (ca) axbycy; the remaining 19 are expressible as transvectants of compounds of these four.
F= ai; the Hessian H = (ab) 2 azbx; the quartic i= (ab) 4 axb 2 x; the covariants 1= (ai) 4 ay; T = (ab)2(cb)aybyci; and the invariants A = (ab) 6; B = (ii') 4 .
The Hessian is symbolically (abc) 2 azbzcz = H 3, and for the canonical form (1 +2m 3)xyz-m 2 (x 3 +y 3 +z 3).
When a z and the invariants B and C all vanish, either A or j must vanish; in the former case j is a perfect cube, its Hessian vanishing, and further f contains j as a factor; in the latter case, if p x, ax be the linear factors of i, f can be expressed as (pa) 5 f =cip2+c2ay; if both A and j vanish i also vanishes identically, and so also does f.
From the invariant a2 -2a 1 a 3 -2aoa4 of the quartic the diminishing process yields ai-2a 0 a 21 the leading coefficient of the Hessian of the cubic, and the increasing process leads to a3 -2a 2 a 4 +2a i a 5 which only requires the additional term-2aoa 6 to become a seminvariant of the sextic. A more important advantage, springing from the new form of S2, arises from the fact that if x"-aix n- +a2x n-2.
Denominator factors, that the complete system of the quadratic is composed of the form itself of degree order I, 2 shown by az 2, and of the Hessian of degree order 2, o shown by a2.
Again, for the cubic, we can find A3(z) - -a6z6 1 -az 3.1 -a 2 z 2.1 -a 3 z 3.1 -a4 where the ground forms are indicated by the denominator factors, viz.: these are the cubic itself of degree order I, 3; the Hessian of degree order 2, 2; the cubi-covariant G of degree order 3, 3, and the quartic invariant of degree order 4, o.
By the Prussian Rhine province and the Hessian province of Rhein-Hessen; on the E.
In other directions also the expansion has been rapid; the village of Bornheim was incorporated in Frankfort in 1877, the former Hessian town of Bockenheim in 1895, and the suburbs of Niederrad, Oberrad and Seckbach in 1900.
A third of the leading brigade (British) was killed and wounded in the vain attempt to break through the strong defences of the village, and some French squadrons charged upon it as it retired; a colour was captured in the melee, but a Hessian brigade in second line drove back the cavalry and retook the colour.
The French army, similarly arrayed, but with a few battalions attached to the cavalry wings, was more heterogeneous than the German, being composed of French, Hessian, German mercenaries, and Liegeois.
In 1846 he was appointed professor at Marburg, and though this small university offered little scope for his activities as a teacher, a seat in the Hessian Landtag gave him his first experience of political affairs.
23 (Hamburg, 1853), 226-327 (letters of Hessian deputies); Acta et scripta synodalia Dordracena ministrorum Remonstrantium, Hardervici, 1620 (valuable side-lights); A.
Among other public places and buildings worthy of notice are the Roman Catholic church, with a splendid interior; the Kiinigs-platz, with a remarkable echo; the Karls-platz, with the statue of the landgrave Charles; and the Martins-platz, with a large church - St Martin's - with twin towers, containing the burial-vaults of the Hessian princes.
There are also the Bose Museum, containing collections of pictures and antiquities of Hessian origin, museums of natural history and ethnography, an industrial exhibition hall, and an industrial art school.
On the Hessian fly, Cecidomyia destructor, Say, the May brood of which produces swellings immediately above the joints of barley attacked by it, see Asa Fitch, The Hessian Fly (Albany, 1847), reprinted from Trans.
The inhabitants are mainly of the Saxon stock and speak Low German dialects, except in the Upper Frankish district around Siegen, where the Hessian dialect is spoken.
Among interesting landmarks are the Federal Inn (1763),(1763), in which President Washington was entertained in 1794, and which has been used as a banking house since 1814; the old county gaol (1770), used as such until 1848; and the site of the "Hessian Camp," where some of the prisoners captured during the War of Independence were confined.
(1760-1785), who had become a Roman Catholic, 22,000 Hessian troops were hired by England for about X 3,191,000, to assist in the war against the North American colonies.
In 1821 a small mediate principality was formed out of the old lordship of Ratibor and certain ecclesiastical domains, and was conferred upon Victor Amadeus, landgrave of Hesse-Rotenburg, as compensation for some Hessian territory absorbed by Prussia.
Outside this, though not in hostility to it, Hanover, Brunswick, Oldenburg and Schaumburg-Lippe formed a separate customs-union, (Steuerverein) by treaties signed on the 1st of May 1834 and the 7th of May 1836, and to this certain Prussian and Hessian enclaves were attached.
Prussia, on the other hand, announced its determination to carry out the principles of the Union and to maintain the Hessian constitution.
What this means is that the Hessian lines are managed by the Prussian department.
Chief among these is the barracks, erected by the colony in 1758 to mitigate the evils of billeting, and occupied by British troops during the Seven Years' War, and at different times by British, Hessian and American troops during the War of Independence.
The Hessian star catalogue was published in Lucius Barettus's Historia coelestis (Augsburg, 1668), and a number of other observations are to be found in Coeli et siderum in eo errantium observationes Hassiacae (Leiden, 1618), edited by Willebrord Snell.
In the war between Austria and Prussia in 1866, HesseDarmstadt was upon the side of the Austrians; Prince Louis accompanied his troops to the front, and was duly appointed by the grand-duke to the command of the Hessian division.
The demand for this class of bagging, which is made from fine hessian yarns, is still great.
These fine Rio hessian yarns form an important branch of the Dundee trade, and in some weeks during 1906 as many as 1000 bales were despatched to Brazil, besides numerous quantities to other parts of the world.
Amongst these may be mentioned the following: Hessian, bagging, tarpaulin, sacking, scrims, Brussels carpets, Wilton carpets, imitation Brussels, and several other types of carpets, rugs and matting, in addition to a large variety of fabrics of which jute forms a part.
Secondly, as to the inflections, the process is a similar one; it can be shown that the inflections are the intersections of the curve by a derivative curve called (after Ludwig Otto Hesse who first considered it) the Hessian, defined geometrically as the locus of a point such that its conic polar (§ 8 below) in regard to the curve breaks up into a pair of lines, and which has an equation H = o, where H is the determinant formed with the second differential coefficients of u in regard to the variables (x, y, z); H= o is thus a curve of the order 3 (m - 2), and the number of inflections is =3m(m-2).
Xxviii., 1844); in the latter of these the points of inflection are obtained as the intersections of the curve u = o with the Hessian, or curve A = o, where A is the determinant formed with the second derived functions of u.
We have in the Hessian the first instance of a covariant of a ternary form.
The Hessian A has just been spoken of as a covariant of the form u; the notion of invariants and covariants belongs rather to the form u than to the curve u=o represented by means of this form; and the theory may be very briefly referred to.
Intellectual interests of a high order have always characterized, Leipzig, and what Karl von Holtei once said of it is true to-day: "There is only one city in Germany that represents Germany; only a single city where one can forget that he is a Hessian, a Bavarian, a Swabian, a Prussian or a Saxon; only one city where, amid the opulence of the commercial world with which science is so gloriously allied, even the man who possesses nothing but his personality is honoured and esteemed; only one city, in which, despite a few narrownesses, all the advantages of a great, I may say a world-metropolis, are conspicuous !
The church also contains the tombs of numerous Hessian landgraves and knights of the Teutonic Order.
In Germany, Ips and Passau on the Danube, and Gross Almerode in Hesse, are the best known localities producing fireclay goods, the crucibles from the last-mentioned place, known as Hessian crucibles, going all over the world.
He had just finished dressing for his ride, and wore a blue uniform, opening in front over a white waistcoat so long that it covered his rotund stomach, white leather breeches tightly fitting the fat thighs of his short legs, and Hessian boots.
Men in military uniforms and Hessian boots could be seen through the windows, laughing and walking through the rooms.
He wore a woman's loose gown of frieze, blue trousers, and large torn Hessian boots.
Jauntily shifting the position of his legs in their tight riding breeches, diffusing an odor of perfume, and admiring his partner, himself, and the fine outlines of his legs in their well-fitting Hessian boots, Nicholas told the blonde lady that he wished to run away with a certain lady here in Voronezh.
He first studied theology at Giessen, but after the campaign of 1814, in which, like his brother August, he took part as a Hessian volunteer, began the study of jurisprudence, and in 1818 established himself as Privatdocent of civil law at Giessen.
For a short while he was at the head of the new Hessian administration; but his ambition was to share in the creation of a united Germany.
American literature began as far back as 1788, when a report on the Hessian fly was issued by Sir Joseph Banks; in 1817 Say began his writings; while in 1856 Asa Fitch started his report on the " Noxious Insects of New York."
Hundreds of acres of wheat are lost annually in America by the ravages of the Hessian fly; the fruit flies of Australia and South Africa cause much loss to orange and citron growers, often making it necessary to cover the trees in muslin tents for protection.
It was recovered by the landgrave of Thuringia in 1388, but soon reverted to Hesse, and it became the residence of one of the branches of the Hessian royal house, a branch which died out in 1655.
The best-known dipterous pests are the Hessian fly (Cecidosnyia destructor), the pear midge (Diplosis pyrivora), the fruit flies (Tephritis Tyroni of Queensland and Halterophora capitata or the Mediterranean fruit fly), the onion fly (Phorbia cepetorum), and numerous corn pests, such as the gout fly (Chloropstaeniopus) and the frit fly (Oscinis frit).
The Hessian 0 =A 2 is such that (f, 2 and if f is expressible in the form X(p x) 3 +,i(g x) 3, that is as the sum of two perfect cubes,.
In 1864 he went as Hessian envoy to Vienna, retiring in 1872 when the post was abolished.
We find that Di must be equal to p x g x for then t x (p x) 3 +, u (g x) 3, Hence, if px, qx be the linear factors of the Hessian 64, the cubic can be put into the form A(p x) 3 +ï¿½(g x) 3 and immediately solved.
Many reforms in ecclesiastical, educational, financial and administrative matters were introduced, and in general the grand-duchy may be said to have passed largely under the influence of Prussia, which, by an arrangement made in 1896, controls the Hessian railway system.
In general for a form in n variables the Hessian is 3 2 f 3 2 f a2f ax i ax n ax 2 ax " ï¿½ï¿½ ' axn and there is a remarkable theorem which states that if H =o and n=2, 3, or 4 the original form can be exhibited as a form in I, 2, 3 variables respectively.
The badge is an open oval of gold with the Hessian lion in the centre.