# Vanishes Sentence Examples

- This stream flows north-westward from the last lake and
**vanishes**underground within 3 m. - We find Prester John in one more phase before he
**vanishes**from Asiatic history, real or mythical. - If 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. - If 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. - As 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. - Thus Russell's contradiction
**vanishes**, and an examination of the other contradictions shows that they vanish also. - Since 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 .> - X 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. - =xï¿½ = o is the only solution; but if A
**vanishes**the equations can be satisfied by a system of values other than zeros. - Hence 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. - +(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). - For 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. - To 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. - The first transvectant, (f,f') 1 = (ab) a x b x,
**vanishes**identically. - If 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. - The 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. - This method of solution fails when the discriminant R
**vanishes**, for then the Hessian has equal roots, as also the cubic f. - If, moreover, 0
**vanishes**identically f is a perfect cube. - The quartic has four equal roots, that is to say, is a perfect fourth power, when the Hessian
**vanishes**identically; and conversely. - When C
**vanishes**j has the form j = pxg x, and (f,j) 3 = (ap) 2 (aq)ax = o. - We 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. - When 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. - 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. - For 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. - (18) This
**vanishes**if sin 0 cos 4)=2 sin 4 cos 0, i.e. - +XL
**vanishes**identically, and X is indeterminate. - Cos k(at-r), it is necessary to suppose that the integrated term
**vanishes**at the upper limit. - It
**vanishes**when u =mlr, m being any whole number other than zero. - In 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. - If 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. - The 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. - Hen 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. - But 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.. - If "=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. - If R = 2A, I
**vanishes**at E =o; but the whole illumination, represented by I df, is independent of the value of R. - In 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. - In 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. - When hardened in spirit, however, the greater part of this experimental amyloid in the fowl
**vanishes**, and the reactions are not forthcoming. - If 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. - For 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. - The cilia are lost, the eye-spots disappear, the digestive sac
**vanishes**and the larva becomes a sac or "sporocyst" full of germ-cells. - If 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. - But 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. - The 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. - From these hills southward the ridge gradually becomes less abrupt until in Walsh county it
**vanishes**into prairie. - A 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. - If 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. - The dualism, therefore, between "practice" and "theory" also
**vanishes**; a "theory" unrelated to practice (however indirectly) is simply an illusion. - Anher 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. - If, in particular, the point K coincides with A, so that the resultant
**vanishes**, the given system of forces is said to be FIG. - This
**vanishes**, i.e. - When the latter invariant, but not the former,
**vanishes**, the displacement is equivalent to a pure rotation. - When the latter invariant, but not the former,
**vanishes**, the system reduces to a single force. - It 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. - In the case of a central force, with 0 as pole, the transverse, acceleration
**vanishes**, so that r2dO/dt=h,, (15) - A point on a central orbit where the radial velocity (drfdt)
**vanishes**is called an apse, and the corresponding radius is called an apse-line. - About the vertical; this
**vanishes**if ~ FIG. - If 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. - I feel that I cannot vanish, since nothing
**vanishes**in this world, but that I shall always exist and always have existed.