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The vanishing of this invariant is the condition for equal roots.

3514Locks (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.

176The 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.

88We 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.

751818), 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.

63Direct influence from the Greek Fathers upon the West is vanishing as the Greek language is forgotten.

63This implies the vanishing of the discriminant of the original form.

65This implies the vanishing of the discriminant of the original form.

65The narrow alleys of Porto, Pendino and Mercato have nearly all disappeared, and old Naples has been vanishing day by day.

43The narrow alleys of Porto, Pendino and Mercato have nearly all disappeared, and old Naples has been vanishing day by day.

43The 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 .

44Gordan 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.

44Gordan 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.

44The vanishing of the invariants i and j is the necessary and sufficient condition to ensure the quartic having three equal roots.

45If, however, the condition be the vanishing of i, f contains a linear factor to the fourth power.

45In 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).

46In this case the motion in different directions varies as cos°, vanishing at right angles to the direction of propagation of the primary wave.

34Its proximal centrosome (its distal one vanishing as such).

34the 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", ..

23the 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", ..

23If 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.

24Bidwell 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.

24The 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.

24diminishes, 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.

12Faster 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.

12each 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.

13For 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.

13CY 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.

13= = o, and have in consequence a vanishing resultant.

13When 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.

13On 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.

13(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.

13The place of the vanishing native species is being taken by such European arrivals as sky-larks, finches, blackbirds, sparrows and rooks.

13The 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.

13The 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.

13The 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.

13each 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.

13The 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.

16Two of these show that the leading coefficient of any covariant is an isobaric and homogeneous function of the coefficients of the form; the remaining two may be regarded as operators which cause the vanishing of the covariant.

00Every other concomitant is a rational integral function of these four forms. The linear covariant, obviously the Jacobian of a x and x x is the line perpendicular to x and the vanishing of the quadrinvariant a x is the condition that a x passes through one of the circular points at infinity.

00dp dpu dpv dpw -z)' reducing to the first line, the second line vanishing in consequence of the equation of continuity; and so the equation of motion may be written in the more usual form du du du du d dt +udx+vdy +wdz =X -n dx' with the two others dv dv dv dv i dp dt +u dx +v dy +w dz - Y -P d y' dw dw dw Z w dw i d p dt +u dx +v dy +wd - -P dz.

00He was growing old; the in.doinitable energy of his early career was beginning to slacken; his dreams of extended empire were vanishing.

00Amaze your friends with vanishing eyeballs, angry genies and a dead man's finger.. .

00All the mystery of his vanishing revolved on your description of a hulking fellow in a red neckcloth.

00Ancient, even tribal attachments were not vanishing in the globalized 21st century; on the contrary, they were becoming more intense.

00So long as the particles are small no such vanishing of light in oblique directions is observed, and we are thus led to the conclusion that the hypothesis of a finite AN and of vibrations in the plane of polarization cannot be reconciled with the facts.

00For 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=Ã¯¿½.., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J.

00CY 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.

00= = o, and have in consequence a vanishing resultant.

00The vanishing of this invariant is the condition for equal roots.

00The 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 .

00The vanishing of the invariants i and j is the necessary and sufficient condition to ensure the quartic having three equal roots.

00If 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.

00The simplest invariant is S = (abc) (abd) (acd) (bcd) cf degree 4, which for the canonical form of Hesse is m(1 -m 3); its vanishing indicates that the form is expressible as a sum of three cubes.

00When 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.

00If, however, the condition be the vanishing of i, f contains a linear factor to the fourth power.

00Two of these show that the leading coefficient of any covariant is an isobaric and homogeneous function of the coefficients of the form; the remaining two may be regarded as operators which cause the vanishing of the covariant.

00In 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).

00The 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.

00Every other concomitant is a rational integral function of these four forms. The linear covariant, obviously the Jacobian of a x and x x is the line perpendicular to x and the vanishing of the quadrinvariant a x is the condition that a x passes through one of the circular points at infinity.

002 It was found that the curve showing the relation of transverse changes of dimensions to magnetizing force was similar in general character to the familiar elongation curves, but the signs were reversed; the curve was inverted, indicating at first retraction, which, after passing a maximum and vanishing in a critical field, was succeeded by elongation.

00Bidwell 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.

00The validity of his fundamental position was impaired by the absence of a well-constituted theory of series; the notation employed was inconvenient, and was abandoned by its inventor in the second edition of his Mecanique; while his scruples as to the admission into analytical investigations of the idea of limits or vanishing ratios have long since been laid aside as idle.

00On 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.

00In this case the motion in different directions varies as cosÃ‚°, vanishing at right angles to the direction of propagation of the primary wave.

00(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.

00Locks (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.

00dp dpu dpv dpw -z)' reducing to the first line, the second line vanishing in consequence of the equation of continuity; and so the equation of motion may be written in the more usual form du du du du d dt +udx+vdy +wdz =X -n dx' with the two others dv dv dv dv i dp dt +u dx +v dy +w dz - Y -P d y' dw dw dw Z w dw i d p dt +u dx +v dy +wd - -P dz.

00The 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.

00The place of the vanishing native species is being taken by such European arrivals as sky-larks, finches, blackbirds, sparrows and rooks.

00How the Roman bishopric rose in status till it became the papacy, how the individual popes - in spite of these and similar repulses - advanced steadily on their path, how they succeeded in founding their primacy within the Church, and in re-establishing and maintaining that primacy notwithstanding severe defeats and long periods in which their prestige sank to the vanishing point, is told elsewhere (see Papacy).

00The 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.

00But 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.

00Its proximal centrosome (its distal one vanishing as such).

00The 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.

00The 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.

00St 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.

00The 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.

00The 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.

00We 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.

00To 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.

00His slender prerogatives had gradually dwindled down to vanishing point.

00This 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.

001818), 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.

00He was growing old; the in.doinitable energy of his early career was beginning to slacken; his dreams of extended empire were vanishing.

00Direct influence from the Greek Fathers upon the West is vanishing as the Greek language is forgotten.

00But an insidious change in its once healthy climate had begun to work its decay; the area of cultivated land round it had shrunk to vanishing point, jungle haunted by wild beasts taking its place; and in 1813 its ruin was completed by a sudden change in the course of the Bhagirathi, which formed a new channel 3 m.

00diminishes, 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.

00is 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.

00With strong systems the vanishing of the points is observed with an auxiliary microscope, formed by means of the inner tube.

00Ancient, even tribal attachments were not vanishing in the globalized 21st century; on the contrary, they were becoming more intense.

00Others report the black dog appearing in their homes in the dark and vanishing when they turn on a light.

00Create combos of vanishing jewels and play to your heart's content.

00The founder of the modern-day fighting game is showing no signs of vanishing any time soon.

00Unfortunately, thick, greasy ointments seem to work better than vanishing creams, lotions, or liquids.

00No matter what year it is, get ready to welcome in the tentative warmth of the season while maintaining the ability to cover up if the cold air comes back to visit before vanishing completely.

00The most common urban legend of all is the story of the vanishing hitchhiker.

00One great resource is the celebrity tattoo gallery at Vanishing Tattoos.

00Vanishing Tattoo: Warning, you will see part of Williams' bottom in order to show off his upper hip tattoo, but nothing too revealing.

00According to Vanishing Tattoo, this is one of the most popular tattoos of all time-even though the original was burned in 1729 (sketches survived).

00Also according to Vanishing Tattoo, there's a story about the hands Dürer drew.

00Clearasil Daily Clear Vanishing Acne Treatment Cream: For those looking for a stronger level of benzoyl peroxide, this treatment contains 10% benzoyl peroxide and bentonite to soak up excess oils.

00We all know the frustration of that beloved fragrance vanishing only moments after we've sprayed it on.

00is 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.

01With strong systems the vanishing of the points is observed with an auxiliary microscope, formed by means of the inner tube.

01The 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.

02The 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.

02But 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.

02St 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.

02To 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.

02His slender prerogatives had gradually dwindled down to vanishing point.

02This 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.

02Lawless 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.

03Lawless 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.

03

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