The vanishing of this invariant is the condition for equal roots.
Bidwell 2 accordingly found upon trial that the Wiedemann twist of an iron wire vanished when the magnetizing force reached a certain high value, and was reversed when that value was exceeded; he also found that the vanishing point was reached with lower values of the magnetizing force when the wire was stretched by a weight.
Each of them satisfied by a common system of values; hence the equation R =o is derived on this supposition, and the vanishing of R expresses the condition that the equations can be satisfied by a common system of values assigned to the variables.
For if u, v, w be the polynomials of orders m, n, p respectively, the Jacobian is (u 1 v 2 w3), and by Euler's theorem of homogeneous functions xu i +yu 2 +zu 3 = mu xv1 +yv2 +zv3 = /IV xw 1+y w 2+ zw 3 = pw; denoting now the reciprocal determinant by (U 1 V2 W3) we obtain Jx =muUi+nvVi+pwWi; Jy=ï¿½.., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J.
CY The proof being of general application we may state that a system of values which causes the vanishing of k polynomials in k variables causes also the vanishing of the Jacobian, and in particular, when the forms are of the same degree, the vanishing also of the differential coefficients of the Jacobian in regard to each of the variables.
The discriminant, whose vanishing is the condition that f may possess two equal roots, has the expression j 2 - 6 i 3; it is nine times the discriminant of the cubic resolvent k 3 - 2 ik- 3j, and has also the expression 4(1, t') 6 .
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 vanishing of the invariants i and j is the necessary and sufficient condition to ensure the quartic having three equal roots.
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.
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.
If, however, the condition be the vanishing of i, f contains a linear factor to the fourth power.
In order to obtain the seminvari ants we would write down the (w; 0, n) terms each associated with a literal coefficient; if we now operate with 52 we obtain a linear function of (w - I; 8, n) products, for the vanishing of which the literal coefficients must satisfy (w-I; 0, n) linear equations; hence (w; 8, n)-(w-I; 0, n) of these coefficients may be assumed arbitrarily, and the number of linearly independent solutions of 52=o, of the given degree and weight, is precisely (w; 8, n) - (w - I; 0, n).
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.
In this case the motion in different directions varies as cos°, vanishing at right angles to the direction of propagation of the primary wave.
Locks (East London Advertiser, 1902); Philip Norman, London vanished and vanishing (1905); Records of the London Topographical Society; Monographs of the Committee for the Survey of the Memorials of Greater London.
On either side of any one of them the illumination is distributed according to the same law as for the central image (m = o), vanishing, for example, when the retardation amounts to (mn t 1)X.
(16), secondary disturbance corresponding to the element dS of the plane may be supposed to be that caused by a force of the above magnitude acting over dS and vanishing elsewhere; and it only remains to examine what the result of such a force would be.
The place of the vanishing native species is being taken by such European arrivals as sky-larks, finches, blackbirds, sparrows and rooks.
Its proximal centrosome (its distal one vanishing as such).
We find that if the series of excitations of the muscle be prolonged beyond the short stage of initial improvement, the contractions, after being well maintained for a time, later decline in force and speed, and ultimately dwindle even to vanishing point.
This necessitates the satisfying of five equations; in other words, there are five alterations of the 3rd order, the vanishing of which produces an image of the 5th order.
The narrow alleys of Porto, Pendino and Mercato have nearly all disappeared, and old Naples has been vanishing day by day.
1818), who was more than any other man both the representative of an epoch fast vanishing and the harbinger of the new spirit that was stirring young Rumania.
Direct influence from the Greek Fathers upon the West is vanishing as the Greek language is forgotten.
The second and third terms each vanishing, it is a, b, c a', c', , a", bn c",, But if the linear equations hold good, then the first column of the 1 The expression, a linear function, is here used in its narrowest sense, a linear function without constant term; what is meant is that the determinant is in regard to the elements a, a', a", ..
Diminishes, vanishing and changing sign when the temperature of the hot junction is nearly as much above the neutral point as the temperature of the cold junction is below it.
Is projected depends upon the relation of the "vanishing line" to the circle; if it intersects it in real points, then the projection is a hyperbola, if in imaginary points an ellipse, and if it touches the circle, the projection is a parabola.
With strong systems the vanishing of the points is observed with an auxiliary microscope, formed by means of the inner tube.
Faster and faster, vying with one another, they moved at the double or at a trot, vanishing amid the clouds of dust they raised and making the air ring with a deafening roar of mingling shouts.
The stars, as if knowing that no one was looking at them, began to disport themselves in the dark sky: now flaring up, now vanishing, now trembling, they were busy whispering something gladsome and mysterious to one another.
= = o, and have in consequence a vanishing resultant.
This implies the vanishing of the discriminant of the original form.
The Latin element in Africa and the Christian faith almost disappeared in a single generation; 1 the Berbers of the [1 The North African Church was not utterly swept away by the Moslem conquest, though its numbers at that time were very greatly diminished, and thereafter fell gradually to vanishing point, partly by emigration to Europe.
The totality of being may thus be conceived as a series of concentric circles, fading away towards the verge of non-existence, the force of the original Being in the outermost circle being a vanishing quantity.
But in the actual phenomenal world unity and harmony are replaced by strife and discord; the result is a conflict, a becoming and vanishing, an illusive existence.
The height and massiveness of the mountains decrease to the south-west, where the piedmont belt sweeps westward around them in western Georgia and eastern Alabama Some of the residual mountains hereabouts are reduced to a mere skeleton or framework by the retrogressive penetration of widening valleys between wasting spurs; the very type of vanishing forms, Certain districts within the mountains, apparently consisting of less resistant crystalline rocks, have been reduced to basin-like peneplains in the same time that served only to grade the slopes and subdue the summits of the neighboring mountains of more resistant rocks; the best example of this kind is the Asheville peneplain in North Carolina, measuring about 40 by 20 m.
The disclosures before the Parnell Commission, the O'Shea divorce proceedings, the downfall of Mr Parnell and the disruption of the Irish party, assisted him in his task; but the fact remains that by persistent courage and undeviating thoroughness he reduced crime in Ireland to a vanishing point.
St Benedict introduced too into the monastic life the idea of law and order, of rule binding on the abbot no less than on the monks; thus he reduced almost to a vanishing point the element of arbitrariness, or mere dependence on the abbot's will and whim, found in the earlier rules.
The university, founded in 1389 by the sole efforts of the citizens, soon gained a great reputation; in the 15th century its students numbered much more than a thousand, and its influence extended to Scotland and the Scandinavian kingdoms. Its decline began, however, from the moment when the Catholic sentiment of the city closed it to the influence of the Reformers; the number of its students sank to vanishing point, and though, under the influence of the Jesuits, it subsequently revived, it never recovered its old importance.
The export of ships from the United Kingdom to the empire dgcreased during two years, 1903 (~3o5,682) and 904 (~365,o62), almost to a vanishing point, German yards being able to cope with the demands made upon them for the supply of vessels of all classes, including mercantile vessels and ships of war.
To those, of course, for whom the only real identity is identity in difference, while identity without difference, like difference without identity, is simply a limit or a vanishing point, Herbart's logic and metaphysic will alike lack plausibility.
His slender prerogatives had gradually dwindled down to vanishing point.
Lawless and predatory, the English settlers were hardly distinguishable from the native Irish, and the authority of the English king over both had been reduced to vanishing point.