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We find Prester John in one more phase before he vanishes from Asiatic history, real or mythical.

76This stream flows north-westward from the last lake and vanishes underground within 3 m.

43If all the water be frozen, we have the vapour pressure curve of ice OB; while, if the pressure be raised, so that all the vapour vanishes, we get the curve OC of equilibrium between the pressure and the freezing point of water.

20The effect of each of the elements of the grating is then the same; and, unless this vanishes on account of a particular adjustment of the ratio a: d, the resultant amplitude becomes comparatively very great.

10If this vanishes for all values of X, u, v, ~, s~, ~ we must have X, Y, Z, L, M, N = o, which are the conditions of equilibrium.

10Since the determinant having two identical rows, and an3 an3 ï¿½ï¿½ï¿½ ann vanishes identically; we have by development according to the elements of the first row a21Au+a22Al2 +a23A13+ï¿½ï¿½ï¿½ +a2nAin =0; and, in general, since a11A11+a12A12 +ai 3A13+ï¿½ï¿½ ï¿½ +ainAin = A, if we suppose the P h and k th rows identical a A +ak2 A 12 +ak3A13+ï¿½ï¿½ï¿½ +aknAin =0 (k > i) .and proceeding by columns instead of rows, a li A lk +a21A2k + a 31A3k+ï¿½ï¿½ï¿½+aniAnk = 0 (k .>

11As regards the polarization of the dispersed light as dependent on the angle at which it is emitted, we find that although, when terms of the second order are included, the scattered light no longer vanishes in the same direction as before, the peculiarity is not lost but merely transferred to another direction.

00Thus Russell's contradiction vanishes, and an examination of the other contradictions shows that they vanish also.

00x n / I l yl, y2,ï¿½ï¿½ï¿½yn Theorem.-If the functions y 1, y2,ï¿½ï¿½ï¿½ y n be not independent of one another the functional determinant vanishes, and conversely if the determinant vanishes, yl, Y2, ...y.

00Hence in all there are mn such systems. If, therefore, we have a third equation, and we substitute each system of values in it successively and form the product of the mn expressions thus formed, we obtain a function which vanishes if any one system of values, common to the first two equations, also satisfies the third.

00+(m -3) D 5(213) (214) (15) - (13) (14) (14), as= and and we see further that (alai +a2a2+...+amam) k vanishes identically unless (mod m).

00For a single quantic of the first order (ab) is the symbol of a function of the coefficients which vanishes identically; thus (ab) =a1b2-a2bl= aw l -a1ao=0 and, indeed, from a remark made above we see that (ab) remains unchanged by interchange of a and b; but (ab), = -(ba), and these two facts necessitate (ab) = o.

00This will be recognized as the resultant of the two linear forms. If the two linear forms be identical, the umbral sets a l, a2; b l, b 2 are alternative, are ultimately put equal to one another and (ab) vanishes.

00To obtain the corresponding theorem concerning the general form of even order we multiply throughout by (ab)2' 2c272 and obtain (ab)2m-1(ac)bxc2:^1=(ab)2mc2 Paying attention merely to the determinant factors there is no form with one factor since (ab) vanishes identically.

00The first transvectant, (f,f') 1 = (ab) a x b x, vanishes identically.

00If the covariant (f,4) 1 vanishes f and 4 are clearly proportional, and if the second transvectant of (f, 4 5) 1 upon itself vanishes, f and 4) possess a common linear factor; and the condition is both necessary and sufficient.

00In this case (f, ï¿½) 1 is a perfect square, since its discriminant vanishes.

00The discriminant of f is equal to the discriminant of 0, and is therefore (0, 0') 2 = R; if it vanishes both f and 0 have two roots equal, 0 is a rational factor of f and Q is a perfect cube; the cube root being equal, to a numerical factor pres, to the square root of A.

00This method of solution fails when the discriminant R vanishes, for then the Hessian has equal roots, as also the cubic f.

00If, moreover, 0 vanishes identically f is a perfect cube.

00The quartic has four equal roots, that is to say, is a perfect fourth power, when the Hessian vanishes identically; and conversely.

00When C vanishes j has the form j = pxg x, and (f,j) 3 = (ap) 2 (aq)ax = o.

00The discriminant is the resultant of ax and ax and of degree 8 in the coefficients; since it is a rational and integral function of the fundamental invariants it is expressible as a linear function of A 2 and B; it is independent of C, and is therefore unaltered when C vanishes; we may therefore take f in the canonical form 6R 4 f = BS5+5BS4p-4A2p5.

00We can see that (abc)a x b x c x is not a covariant, because it vanishes identically, the interchange of a and b changing its sign instead of leaving it unchanged; but (abc) 2 is an invariant.

00When R =0, and neither of the expressions AC - B 2, 2AB -3C vanishes, the covariant a x is a linear factor of f; but, when R =AC - B 2 = 2AB -3C =0, a x also vanishes, and then f is the product of the form jx and of the Hessian of jx.

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.

00for 0=2, (a l a i +v 2 a 2) w; either v l or cr 2 will vanish if a1a2=A2=o; but every term, in the development, is of the form (222...)Ar and therefore vanishes; so that none are left to undergo reduction.

00(18) This vanishes if sin 0 cos 4)=2 sin 4 cos 0, i.e.

00For strong magnetizing forces (which in these experiments did not exceed II= 48.9) the permeability remains almost constant at its initial value (about 400), until the temperature is within nearly i oo of the critical point; then the permeability diminishes more and more rapidly until the critical point is reached and the magnetization vanishes.

00+XL vanishes identically, and X is indeterminate.

00cos k(at-r), it is necessary to suppose that the integrated term vanishes at the upper limit.

00It vanishes when u =mlr, m being any whole number other than zero.

00In like manner we may find the illumination in any other direction, and it is obvious that it vanishes when sin 0 is any multiple of A/a.

00If we take a direction such that the light (of given wave-length) from a single aperture vanishes, the evanescence continues even when the whole series of apertures is brought into contemplation.

00hen d =a the general formula becomes sin' Zm7r Bm: B = (3), showing that, when m is even, B m vanishes, and that, when m is odd, B m: B =1/m272.

00But if we now suppose that Q lies on the circle u= a cos 0, the middle term vanishes, and we get, correct as far as w4, QP= (u+a sin 4) sin w) 1 ' 3 1 {- a sin2c?sin4w V 4u so that QP - u=asin0sinw -Ft asin¢tanOsin 4 w..

00If "=4), the term of the first order vanishes, and the reduction of the difference of path via P and via A to a term of the fourth order proves not only that Q and Q' are conjugate foci, but also that the foci are exempt from the most important term in the aberration.

00At a moment when the eye, or object-glass of a telescope, occupies a dark position, the star vanishes.

00If R = 2A, I vanishes at E =o; but the whole illumination, represented by I df, is independent of the value of R.

00In the second term if we observe that cos {p'+ 27rh/Af)E} =cos{p' - g,E} = cos p cos g, +sin p sin giE, we see that the second part vanishes when integrated, and that the remaining integral is of the form w = f +.0 sin z h, cos where h,=7rh/Af, g,=a-27Th/Af.

00In the limiting case in which the medium is regarded as absolutely incompressible S vanishes; but, in order that equations (2) may preserve their generality, we must suppose a at the same time to become infinite, and replace a 2 3 by a new function of the co-ordinates.

00When hardened in spirit, however, the greater part of this experimental amyloid in the fowl vanishes, and the reactions are not forthcoming.

00If w vanishes throughout the fluid at any instant, equation (I I) shows that it will always be zero, and the fluid motion is then called irrotational; and a function 4) exists, called the velocity function, such that udx+vdy-{-wdz =-d, (13) and then the velocity in any direction is the space-decrease or downward gradient of cp.

00For a doublet at S, of moment m, the Stokes' function is M f cos PSx = - m p s3; and for its image at H the Stokes' function is m f cos PHx =m f 3 PH" (6) so that for the comnation _ a3 I I 2 4)-myb12 (f 3 PH PS 3) =m f 3 (pa ll 3 P53)' 3 and this vanishes over the surface of the sphere.

00The cilia are lost, the eye-spots disappear, the digestive sac vanishes and the larva becomes a sac or "sporocyst" full of germ-cells.

00If there is no external world, the distinction between substance and accidents vanishes, and these become the sole essence of material objects, so that there is no room for any change whilst they remain as before.

00But now that the date is put back to about 112 the difficulty vanishes, since Polycarp was not much over forty when he received the letter.

00The term Pv/V added up for a complete wave vanishes, for P/V is constant and Zv=o, since on the whole the compression equals the extension.

00From these hills southward the ridge gradually becomes less abrupt until in Walsh county it vanishes into prairie.

00A few of the large streams may, when in flood, spr.ead out in a temporary shallow sheet qn a dead level of clay, or playa, in a basin centre, but the sheet of water vanishes in the warm season and the stream shrinks far up its course, the absolutely barren clay floor of the playa, impassable when wet, becomes firm enough for crossing when dry.

00If we measure the osmotic pressure Po when the solvent is under its own vapour pressure only, that is, when P = p = Po, the term involving V vanishes, and the limit of integration P' becomes Pod-p. If we assume that V', the volume change on dilution, varies regularly or not appreciably with pressure, we may write the first integral as V' (P o -{- p - p') where V' now denotes its mean value between the limits.

00The dualism, therefore, between "practice" and "theory" also vanishes; a "theory" unrelated to practice (however indirectly) is simply an illusion.

00Anher appears in the XIth dynasty; and Khentamenti, the god of the western Hades, rises to importance in the middle kingdom and then vanishes in the XVIIIth.

00But the numerical factor appears to be yz'+zy', while it is the quantity yz' - zy' which really vanishes.

00If, in particular, the point K coincides with A, so that the resultant vanishes, the given system of forces is said to be FIG.

00This vanishes, i.e.

00When the latter invariant, but not the former, vanishes, the displacement is equivalent to a pure rotation.

00When the latter invariant, but not the former, vanishes, the system reduces to a single force.

00It appears from (24) that through any assigned point 0 three rectangular axes can be drawn such that the product of inertia with respect to each pair of co-ordinate planes vanishes; these are called the principal axes of inertia at 0.

00In the case of a central force, with 0 as pole, the transverse, acceleration vanishes, so that r2dO/dt=h,, (15)

00A point on a central orbit where the radial velocity (drfdt) vanishes is called an apse, and the corresponding radius is called an apse-line.

00about the vertical; this vanishes if ~ FIG.

00If at any instant the direction of one of the crank-arms coincides with the line of connection, the common perpendicular of the line sf connection and the axis of that crank-arm vanishes, and the 3irectional relation of the motions becomes indeterminate.

00The latter tendency is called a centrifugal couple, and vanishes for rotation about a principal axis.

00Again, the form is such that if it be taken away the nature infallibly vanishes...

00The story of James de la Cloche is indeed itself another historical mystery; he abruptly vanishes as such at Rome at the end of 1668, and thus provides a disappearance of convenient date; but the question concerning him is complicated by the fact that a James Henry de Bovere Roano Stuardo, who married at Naples early in 1669 and undoubtedly died in the following August, claiming to be a son of Charles II., makes just afterwards an equally abrupt appearance; in many respects the two men seem to be the same, but Monsignor Barnes, following Lord Acton, here regards James Stuardo as an impostor who traded on a knowledge of James de la Cloche's secret.

00If the prayer for vengeance begins here, Budde's " deep division in the middle of an acrostic letter-group " vanishes.

00So, with the collapse of the empire, the imperial art vanishes also: and when, some 500 years later, a new art arose under the Sassanids, whose achievements stand to those of Achaemenid art in much the same relation as the achievements of the two dynasties to each other, we discover only isolated reminiscences of its predecessor.

00Henceforward Greek culture practically vanishes and gives place to Aramaic; it is significant that in future the kings of Mesene stamped their coinage with Aramaic legends.

00But the most characteristic passage of the epopee is the mysterious disappearance of Shah Kaikhosrau, who suddenly, when at the height of earthly fame and splendour, renounces the world in utter disgust, and, carried away by his fervent longing for an abode of everlasting tranquillity, vanishes for ever from the midst of his companions.

00Since 0(o) is finite, proportional to K, the integrated term vanishes at both limits, and we have simply f 0(z)dz f: (z)dz, (34) and T= ref: z1,1,(z)dz (35) In Laplace's notation the second member of (34), multiplied by 27r, is represented by H.

00The surface-tension diminishes as the temperature rises, and when the temperature reaches that of the critical point at which the distinction between the liquid and its vapour ceases, it has been observed by Andrews that the capillary action also vanishes.

00Charles, duke of Lower Lorraine, who was thrown into prison by Hugh Capet in 991, left two sons, the last male descendants of the Carolingians, Otto, who was also duke of Lower Lorraine and died without issue, and Louis, who after the year loon vanishes from history.

00For though Aristotle's divergence from Plato is very conspicuous when we consider either his general conception of the subject of ethics, or the details of his system of virtues, still his agreement with his master is almost complete as regards the main outline of his theory of human good; the difference between the two practically vanishes when we view them in relation to the later controversy between Stoics and Epicureans.

00The two systems that have just been described were those that most prominently attracted the attention of the ancient world, so far as it was directed to ethics, from their Later almost simultaneous origin to the end of the 2nd Greek century A.D., when Stoicism almost vanishes from our philo- view.

00broken up into a sum of (3 3 =) 2 7 determinants, each of which is either of some such form as a, a, a' a' b' a", a", where the term a/3y' is not a term of the a/3y-determinant, and its coefficient(as a determinant with two identical columns) vanishes; or else it is of a form such as t af'y".

00In certain cases, indeed, one portion vanishes entirely: thus the stream ordinarily refracted in the first rhomb gives an ordinary or an extraordinary stream alone in the second, according as the principal planes are parallel or perpendicular, the reverse being the case with the extraordinary stream of the first rhomb.

00the hot junction is raised to 270 C., at which temperature the thermoelectric power vanishes and the metals are said to be neutral to one another.

00It would appear, for instance, according to the observations of Dewar and Fleming, that the value of s for iron is positive below - 150° C., at which point it vanishes.

00One of these, called the "Bourne," bursts forth a short way above the town at irregular intervals of one to ten years or more; and after running a torrent for two or three months, as quickly vanishes.

005 Lanval 6 is a fairy story, and the hero vanishes eventually with his fairy princess to the island of Avallon or Avilion.

00A coin vanishes and then audibly reappears in a folded handkerchief held by a spectator.

00impermanent, monk, ignorance vanishes and knowledge arises.

00thin veneer which vanishes as soon as they take on caring responsibilities.

00trunk of the tree he vanishes.

00vanishes altogether.

00vanishes when you stop trying to make cars burn gasoline OR LPG.

00vanishes in a puff of smoke... .

00vanishes in a flash of fire!

00vanishes in an instant!

00vanishes from the deck, only to appear stuck to a window!

00When Sarah gets to the fountain she vanishes leaving only a small whirlwind in the Shadows.

00We have 62=W12+W22+032=P2 (N) 2 Y 2+P2 (A I - 3) 2 If AN, AD be both finite, we learn from (7) that there is no direction perpendicular to the primary (polarized) ray in which the secondary light vanishes.

00So far as (7) is concerned the alternative supposition that AD vanishes would answer equally well, if we suppose the vibrations to be executed in the plane of polarization; but let us now revert to (5), which gives w 3 = _ PAN y z - = + PAN xy _ PAN z 2 - x2 8 N r 2 N r2' W 2 + N r2 (8) from 0 along which there is no scattered light, - two along the axis According to these equations there would be, in all, six directions of y normal to the original ray, and four (y z = =x) at angles of 45Ã‚° with that ray.

00Since the determinant having two identical rows, and an3 an3 Ã¯¿½Ã¯¿½Ã¯¿½ ann vanishes identically; we have by development according to the elements of the first row a21Au+a22Al2 +a23A13+Ã¯¿½Ã¯¿½Ã¯¿½ +a2nAin =0; and, in general, since a11A11+a12A12 +ai 3A13+Ã¯¿½Ã¯¿½ Ã¯¿½ +ainAin = A, if we suppose the P h and k th rows identical a A +ak2 A 12 +ak3A13+Ã¯¿½Ã¯¿½Ã¯¿½ +aknAin =0 (k > i) .and proceeding by columns instead of rows, a li A lk +a21A2k + a 31A3k+Ã¯¿½Ã¯¿½Ã¯¿½+aniAnk = 0 (k .>

00If the jth column be identical with the i ll ' the determinant A vanishes identically; hence if j be not equal to i, k, or r, a 11 a 21 a31 0 =I alk a2k a3k A11.

00Such a determinant when of uneven degree vanishes, for if we multiply each row by - I we multiply the determinant by (- I) n = -1, and the effect of this is otherwise merely to transpose the determinant so that it reads by rows as it formerly did by columns, an operation which we know leaves the determinant unaltered.

00x n / I l yl, y2,Ã¯¿½Ã¯¿½Ã¯¿½yn Theorem.-If the functions y 1, y2,Ã¯¿½Ã¯¿½Ã¯¿½ y n be not independent of one another the functional determinant vanishes, and conversely if the determinant vanishes, yl, Y2, ...y.

00=xÃ¯¿½ = o is the only solution; but if A vanishes the equations can be satisfied by a system of values other than zeros.

00In this case (f, Ã¯¿½) 1 is a perfect square, since its discriminant vanishes.

00Now, when C = o, clearly (see ante) R 2 j = 6 2 p where p = S +2 B a; and Gordan then proves the relation 6R 4 .f = B65Ã¯¿½5B64p - 4A2p5, which is Bring's form of quintic at which we can always arrive, by linear transformation, whenever the invariant C vanishes.

00+bx 2, every leading coefficient of a simultaneous covariant vanishes by the operation of a+Sib=aoda +alda.2+...+a7,-1d a P+bod b Observe that we may employ the principle of suffix diminution to obtain from any seminvariant one appertaining to a (p-I)i c and a q - I ie, and that suffix augmentation produces a portion of a higher seminvariant, the degree in each case remaining unaltered.

00If Q be on the circle described upon OA as diameter, so that u = a cos 4,, then Q' lies also upon the same circle; and in this case it follows from the symmetry that the unsymmetrical aberration (depending upon a) vanishes.

00Again, considering the systems which start from the P normal ranges, we see that there is a probability p/(P+p) which vanishes in the limit, that a system selected at random from these will be in an abnormal state after a sufficient time.

00If, however, hinges or joints are introduced at the points of contrary flexure, they become necessarily points where the bending moment is zero and ambiguity as to the stresses vanishes.

00This couple vanishes when the axis of rotation is a principal axis at 0, and in no other case (cf.

00The contradiction here is palpable; and at the same time the antithesis of " just " and " good " ultimately vanishes.

00It would appear, for instance, according to the observations of Dewar and Fleming, that the value of s for iron is positive below - 150Ã‚° C., at which point it vanishes.

00I feel that I cannot vanish, since nothing vanishes in this world, but that I shall always exist and always have existed.

00It vanishes in the light of day, but shines in the darkness.

00The flatness problem vanishes because the huge expansion blows the universe up so much that it smooths out irregularities.

00For many women, sex equality is a thin veneer which vanishes as soon as they take on caring responsibilities.

00Within a few feet of the old trunk of the tree he vanishes.

00That affects whether type C becomes uncommon, or vanishes altogether.

00LPG burns cleaner - less pollutants - and the power issue vanishes when you stop trying to make cars burn gasoline OR LPG.

00Then he vanishes in a puff of smoke....

00Lighter Than Air: A borrowed lighter is used for a quick and funny gag, then suddenly vanishes in a flash of fire !

00The spectator again chooses a card and it, too vanishes in an instant !

00A chosen card vanishes from the deck, only to appear stuck to a window !

00Depression that arrives as days (and daylight time) shorten in autumn and winter yet vanishes in the spring and summer when longer days of sunlight resume.

00When one half of the Dual Cube is matched with two cubes of the same color, it vanishes, leaving behind a solid cube of the other color.

00Every time a set of balls vanishes, the meter at the top of the screen fills up.

00Virgo men are known for their good looks and winning smiles, but just when you think you have one of them snared, poof, he vanishes.

00The most commonly reported encounter with Resurrection Mary is that people offer her a ride, and then the woman vanishes while inside the car or when they pass the cemetery.

00Usually, it's a woman that acts strange while in the car, but upon reaching the destination she vanishes.

00=xï¿½ = o is the only solution; but if A vanishes the equations can be satisfied by a system of values other than zeros.

01

The word usage examples above have been gathered from various sources to reflect current and historial usage. They do not represent the opinions of YourDictionary.com.