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ax

ax

ax Sentence Examples

  • He swung the ax once more, now only half a body length away.

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  • He swung the ax again.

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  • She relinquished the ax for the glass of tea.

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  • He swung the ax again.

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  • "That's called an ice ax, or piolet," Ryland answered.

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  • Again she swung the ax, and this time it went half way through the log.

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  • Dean leaped on Shipton, clawing away at the soft snow, pummeling him like an eighth grade schoolyard brawler while Shipton, still clutching his ice ax in one hand, swung at Dean, catching him on the cheek and face with the side of the solid handle.

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  • Shipton flailed out at him with his ax, missing his head by inches as Dean leaned sideways and frantically fumbled with his line to drop again.

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  • Shipton flailed out at him with his ax, missing his head by inches as Dean leaned sideways and frantically fumbled with his line to drop again.

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  • l ax 2 2 ax i l This is called the kth transvectant of f over 4); it may be conveniently denoted by (f, (15)k.

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  • He swung his ice ax into the wall in front of him, dug in the toes of his crampons and began to ascend toward Dean.

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  • Shipton's ax bit the ice scarcely a foot below Dean as the man glared up at him, a snarl on his face.

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  • It may be written in the form n n-1 2 ax 1 +bx1 x2 +cx 1 x 2 + ...; or in the form n n n=1 n n-2 2 +(1)bx x2+ ?

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  • I suppose you've got a good reason why you tried to beat the brains out of a guy holding an ice ax, in the middle of the street with a bunch of people watching.

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  • Shipton swung his ice ax again, inching up closer to Dean.

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  • His muscular back glistened with perspiration as he swung the ax, expertly splitting a chunk of wood.

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  • He lifted the ax, taking aim at a new block of wood.

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  • The two forms ax, bx, or of, 0, may be identical; we then have the kth transvectant of a form over itself which may, or may not, vanish identically; and, in the latter case, is a covariant of the single form.

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  • The leg wound from Shipton's flailing ice ax had been an eight-stitcher of no permanent consequence.

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  • 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.

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  • This time the ax sank about four inches into the wood - in another spot.

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  • After a full minute of tugging and grunting she managed to dislodge the ax from the wood.

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  • Such curves are given by the equation x 2 - y 2 = ax 4 -1bx 2 y 2 +cy 4 .

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  • - 2 ay2' ax, Ux2 ��� a y n ay.

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  • Taking two of the equations ax + +cz) x"' 1 +...

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  • Other forms are n-1 n-2 2 ax +nbx x +n(n-i)cx x +..., 1121 2 the binomial coefficients C) being replaced by s!(e), and n 1, n-1 1 n-2 2 ax 1 +l i ox l 'x 2 + L ?cx 1 'x2+..., the special convenience of which will appear later.

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  • As between the original and transformed quantic we have the umbral relations A1 = A1a1 d-A2a2, A2 = /21a1+/22a2, and for a second form B1 =A 1 b 1+ A 2 b 2, B 2 =/21bl +�2b2� The original forms are ax, bi, and we may regard them either as different forms or as equivalent representations of the same form.

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  • So they filled a small boat with the things that he would need the most--an ax, a hoe, a kettle, and some other things.

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  • He hefted the ax to his shoulder and grinned down at her.

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  • The �th polar of ax with regard to y is n-� a aye i.e.

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  • The complete covariant and contravariant system includes no fewer than 34 forms; from its complexity it is desirable to consider the cubic in a simple canonical form; that chosen by Cayley was ax 3 +by 3 + cz 3 + 6dxyz (Amer.

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  • Hit him with an ax, eh!...

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  • Glancing over his shoulder at his advancing pursuer, he knew he'd have to drop far enough and rapidly enough to pass Shipton before the killer could swing out with his deadly ax.

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  • He shrugged and offered her the ax.

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  • The number of partitions of a biweight pq into exactly i biparts is given (after Euler) by the coefficient of a, z xPy Q in the expansion of the generating function 1 - ax.

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  • First observe that with f x =a: = b z = ���,f1 = a l a z ', f 2 = a 2 az-', f x =f,x i +f 2 x i, we find (ab) - (a f) bx - (b f) ax.

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  • The Binary Quadratic.-The complete system consists of the form itself, ax, and the discriminant, which is the second transvectant of the form upon itself, viz.: (f, f') 2 = (ab) 2; or, in real coefficients, 2(a 0 a 2 a 2 1).

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  • =y la xl -i-y2a x2 must also vanish for the root a, and thence ax, and a must also vanish for the same root; which proves that a is a double root of f, and f therefore a perfect square.

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  • If a, a be the linear forms, above defined, he raises the identity ax(0) =ax(aJ3) - (3x(aa) to the fifth power (and in general to the power n) obtaining (aa) 5 f = (a13) 5 az - 5 (a0) 4 (aa) ax?3 -F...

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

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  • Hence, from the identity ax (pq) = px (aq) -qx (ap), we obtain (pet' = (aq) 5px - 5 (ap) (aq) 4 pxg x - (ap) 5 gi, the required canonical form.

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  • The 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.

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  • Thus what have been called seminvariants are not all of them invariants for the general substitution, but are invariants for the particular substitution xl = X11 + J-s12, X 2 = 112 Again, in plane geometry, the most general equations of substitution which change from old axes inclined at w to new axes inclined at w' =13 - a, and inclined at angles a, l3 to the old axis of x, without change of origin, are x-sin(wa)X+sin(w -/3)Y sin w sin ' _sin ax y sin w a transformation of modulus sin w' sin w' The theory of invariants originated in the discussion, by George Boole, of this system so important in geometry.

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  • then of course (AB) = (ab) the fundamental fact which appertains to the theory of the general linear substitution; now here we have additional and equally fundamental facts; for since A i = Xa i +,ia2, A2= - �ay + X a2, AA =A?-}-A2= (X2 +M 2)(a i+ a z) =aa; A B =AjBi+A2B2= (X2 +, U2)(albi+a2b2) =ab; (XA) = X i A2 - X2 Ai = (Ax i + /-Lx2) (- /-jai + Xa2) - (- / J.x i '+' Axe) (X a i +%Ga^2) = (X2 +, u 2) (x a - = showing that, in the present theory, a a, a b, and (xa) possess the invariant property.

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  • (ab), aa, ab, (xa), ax, xx.

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  • To assist us in handling the symbolic products we have not only the identity (ab) cx + (bc) a x + (ca) bx =0, but also (ab) x x+ (b x) a + (ax) b x = 0, (ab)a+(bc)a s +(ca)a b = 0, and many others which may be derived from these in the manner which will be familiar to students of the works of Aronhold, Clebsch and Gordan.

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  • There is no linear covariant, since it is impossible to form a symbolic product which will contain x once and at the same time appertain to a quadratic. (v.) is the Jacobian; geometrically it denotes the bisectors of the angles between the lines ax, or, as we may say, the common harmonic conjugates of the lines and the lines x x .

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  • The linear invariant a s is such that, when equated to zero, it determines the lines ax as harmonically conjugate to the lines xx; or, in other words, it is the condition that may denote lines at right angles.

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  • 3) by F, and its components parallel to the co-ordinate axes by X and Y, we have X= - ax = M(3 cos' 0 - I), Y= - y = M (3 sin 0 cos 0.

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  • Thus, to solve the equation ax e +bx+c = o, we consider, not merely the value of x for which ax 2 +bx+c is o, but the value of ax e +bx+c for every possible value of x.

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  • In this they were completely successful, for they obtained general solutions for the equations ax by = c, xy = ax+by+c (since rediscovered by Leonhard Euler) and cy 2 = ax e + b.

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  • A particular case of the last equation, namely, y 2 = ax e + 1, sorely taxed the resources of modern algebraists.

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  • Denoting them by x, y, so that AB is axis of y and a perpendicular through A the axis of x, and rationalizing (26), we have 2 ax 2 - V 2 Xy 2 - V 2 aAy = o, which represents a hyperbola with vertices at 0 and A.

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  • Ignoring temperature effect, and taking the density as a function of the pressure, surfaces of equal pressure are also of equal density, and the fluid is stratified by surfaces orthogonal to the lines of force; n ap, dy, P d z, or X, Y, Z (4) are the partial differential coefficients of some function P, =fdplp, of x, y, z; so that X, Y, Z must be the partial differential coefficients of a potential -V, such that the force in any direction is the downward gradient of V; and then dP dV (5) ax + Tr=0, or P+V =constant, in which P may be called the hydrostatic head and V the head of potential.

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  • z) = I (ax - I d ?

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  • (2) If the actual motion at any instant is supposed to be generated instantaneously from rest by the application of pressure impulse over the surface, or suddenly reduced to rest again, then, since no natural forces can act impulsively throughout the liquid, the pressure impulse W satisfies the equations I do = I d i dos - ax -u, - - y = -v, Pdz = -t, a =p4)-}-a constant, (4) and the constant may be ignored; and Green's transformation of the energy T amounts to the theorem that the work done by an impulse is the product of the impulse and average velocity, or half the velocity from rest.

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  • u, du +v, du +w, du =o,...,..., (9) ax '?

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  • Uniplanar motion alone is so far amenable to analysis; the velocity function 4 and stream function 1G are given as conjugate functions of the coordinates x, y by w=f(z), where z= x +yi, w=4-Plg, and then dw dod,y az = dx + i ax - -u+vi; so that, with u = q cos B, v = q sin B, the function - Q dw u_vi=g22(u-}-vi) = Q(cos 8+i sin 8), gives f' as a vector representing the reciprocal of the velocity in direction and magnitude, in terms of some standard velocity Q.

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  • Thus if T is expressed as a quadratic function of U, V, W, P, Q, R, the components of momentum corresponding are dT dT dT (I) = dU + x2=dV, x3 =dW, dT dT dT Yi dp' dQ' y3=dR; but when it is expressed as a quadratic function of xi, 'x2, x3, yi, Y2, Y3, U = d, V= dx, ' w= ax dT Q_ dT dT dy 1 dy2 dy The second system of expression was chosen by Clebsch and adopted by Halphen in his Fonctions elliptiques; and thence the dynamical equations follow X = dt x2 dy +x3 d Y = ..., Z ..., (3) = dt1 -y2?y - '2dx3+x3 ' M =..

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  • The Diophantine analysis was a favourite subject with Pell; he lectured on it at Amsterdam; and he is now best remembered for the indeterminate equation ax 2 +1 = y 2, which is known by his name.

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  • (b) - 49(a), where 4)(x) is any function of x, by [c P(x)]; the area of the trapezette whose bounding ordinates are uo and u m may then be denoted by [Ax.

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  • - 2 To show that the area of a cross-section of a - prismoid is of the form ax e -{- bx -{- c, where x is the distance of the section from one end, we may proceed as in § 27.

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  • Rods of different materials may be used as sounders in a Kundt's dust tube, and their Young's moduli may be compared, since: length of rod Then dO U = - ax dx or dt = - UK.

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  • Now µx 2 is very small compared with Ax, so that x is nearly equal to F/X, and as an approximation, F=Ax+µF 2 /A 2, or x=F/A - µF 2 /A 3.

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  • 67 b, superimposed and divided so that the length AX represents the load AX, the length AB the load AB, the length YX the reaction YX, and so forth.

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  • To prove this let AB, AB' be the tangents from any point on the line AX.

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  • Similarly AB/ 2 = AX 2 - DX 2 +DP' 2.

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  • I (6), ax) are the typical pedal ganglia; they are joined to the cerebropleural ganglia by connectives.

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  • Warm mineral springs of note are found at Ax, Aulus and Ussat.

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  • In the geometry of plane curves, the term parabola is often used to denote the curves given by the general equation a' n x n = ym+n, thus ax= y 2 is the quadratic or Apollonian parabola; a 2 x = y 3 is the cubic parabola, a 3 x = y4 is the biquadratic parabola; semi parabolas have the general equation ax n-1 = yn, thus ax e = y 3 is the semicubical parabola and ax 3 = y 4 the semibiquadratic parabola.

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  • Denoting these limits by Pi, Pa we have P1/W=L1H/HK=sin (aX)/cos (0+X),

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  • It appears, moreover, that if 0 be varied P will be least when L1H is at right angles to KLi, in which case Pi =W sin (aX), corresponding to 0 = X.

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  • If a, b, c be the semi-axes of the Binets ellipsoid of G, the quadratic moment with respect to the plane Xx + ~iy + vz =0 will be M(aX + bu + c2vi), and that with respect to a parallel plane ?.x+uy+vz=P (29)

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  • An important property of the diagram is that if points X and x are taken dividing the link CB and the whole acceleration of B about C, namely, cb in the same ratio, then Ax represents the acceleration of the point X in magnitude and direction; cb is called the acceleration image of the rod.

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  • Timid by nature, aware of his impending doom, and ax times justly dissatisfied with himself, he tries all means of reconciling himself to the idea of suicide.

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  • In 1844 Minas published a work, avowedly from a MS. with the superscription Galenus, entitled FaXrlvo13 Eiaa'yw yi, &aX€Krucii.

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  • 1 O rc Stoke 52 t a5 5 Castle ax?

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  • The only roads are bridle-paths, and one municipal road by the Balira valley, connecting Andorra with the high road to Seo de Urgel and Manresa; but in 1904 France and Spain agreed to build a railway from Ax to Ripoll, which would greatly facilitate traffic.

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  • the figure the cortex has been cut away In the order ALCYONto show the axis, ax, and the longiACEA the colony consists tudinal canals, lc, surrounding it.

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  • m, Mouth; mf, mesenterial filament; ax, axis.

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  • a curve is of the first order, second order, third order, &c., according as it is represented by an equation of the first order, ax+by+c = o, or say (1 x, y, 1) = o; or by an equation of the second order, ax 2 +2hxy+by e +2fy+2gx+c=o, say (*I x, y, 1) 2 =o; or by an equation of the third order, &c.; or what is the same thing, according as the equation is linear, quadric, cubic, &c.

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  • Ax, The axis.

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  • Ax' to Ax el, the four segments of the axis with muscular bands.

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  • Considering the equations ax +by +cz =d, a'x +b'y +c' z =d', a"x+b"y+cnz=d" and proceeding to solve them by the so-called method of cross multiplication, we multiply the equations by factors selected in such a manner that upon adding the results the whole coefficient of y becomes = o, and the whole coefficient of z becomes = o; the factors in question are b'c" - b"c', b"c - be", bc' - b'c (values which, as at once seen, have the desired property); we thus obtain an equation which contains on the left-hand side only a multiple of x, and on the right-hand side a constant term; the coefficient of x has the value a(b'c" - b"c') +a'(b"c - bc") +a'(bc' - b'c), and this function, represented in the form a, b,c, a' b'c', a" b" c" is said to be a determinant; or, the number of elements being 32, it is called a determinant of the third order.

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  • Haxey), is hybrid, Ax being the Celtic uisg, water; ey the Anglo-Saxon for island; and holm the Norse word with the same signification.

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  • In 1907 the sanctuary of Athena "of the Brazen House" (X aX KlocKos) was located on the Acropolis immediately above the theatre, and though the actual temple is almost completely destroyed, fragments of the capitals show that it was Doric in style, and the site has produced the longest extant archaic inscription of Laconia, numerous bronze nails and plates and a considerable number of votive offerings, some of them of great interest.

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  • line K'X' parallel to KX such that AX = A'X', then the same curve will be described if we regard K'X' and S' as the given directrix and focus, the eccentricity remaining the same.

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  • In analytical geometry, r the equation axe+2hxy+bye+2gx+2fy+ c = o represents an ellipse when ab > h 2; if the centre of the curve be the origin, the equation is a 1 x 2 +2h 1 xy+b i y 2 =C 1, and if in addition a pair of conjugate diameters are the axes, the equation is further simplified to Ax e +By 2 = C. The simplest form is x 2 /a 2 +y 2 /b 2 = 1, in which the centre is the origin and the major and minor axes the axes of co-ordinates.

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  • Delcass, the French minister for foreign affairs, and on the I 8th of August r904 a convention was signed providing for the construction of (1) the Huesca-Oloron line, (2) a line from Ax les Thermes in the Arige to Ripoll in Catalonia, (3) a line from St Girons in the Arige to Sort, and thence to Lrida.

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  • AX - LES - THERMES, a watering place of south-western France, in the department of Ariege, at the confluence of the Ariege with three tributaries, 26 m.

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  • Diagrammatic longitudinal section of cone, showing the axis (ax) bearing the bracts (br) with peltate sporangiophores (sp) springing from their axils; sm, sporangia.

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  • Part of cone, showing the axis (ax) bearing peltate sporangiophores (sp) without bracts; sm, sporangia.

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  • Diagrammatic longitudinal section of the cone, showing the axis (ax) bearing alternate whorls of bracts (br) and peltate sporangiophores (sp) with their sporangia (sm).

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  • cy, Stele of axis (Ax).

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  • The dorsal segments are sterile, corresponding to the bracts of Sphenophyllum Dawsoni, while the ventral segments constitute peltate sporangiophores, each bearing four sporangia, just as in a ax FIG.

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  • ax, Axis.

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  • angia, usually of very large ax, Axis, bearing the sporophylls (sph), size compared with those of on each of which a sporangium most recent Lycopods, have (sm) is seated.

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  • "That's called an ice ax, or piolet," Ryland answered.

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  • Dean leaped on Shipton, clawing away at the soft snow, pummeling him like an eighth grade schoolyard brawler while Shipton, still clutching his ice ax in one hand, swung at Dean, catching him on the cheek and face with the side of the solid handle.

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  • "The bastard had an ice ax in his hand when I slugged him," Dean answered.

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  • I suppose you've got a good reason why you tried to beat the brains out of a guy holding an ice ax, in the middle of the street with a bunch of people watching.

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  • Penny said, glaring at the officer as she tapped the handle of a sinister looking ice ax.

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  • He swung his ice ax into the wall in front of him, dug in the toes of his crampons and began to ascend toward Dean.

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  • Shipton swung his ice ax again, inching up closer to Dean.

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  • He swung the ax once more, now only half a body length away.

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  • Glancing over his shoulder at his advancing pursuer, he knew he'd have to drop far enough and rapidly enough to pass Shipton before the killer could swing out with his deadly ax.

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  • Shipton's ax bit the ice scarcely a foot below Dean as the man glared up at him, a snarl on his face.

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  • Shipton continued to chop, as if deciding this and not a direct blow from the ice ax was a far better way to remove this annoying impediment to his foolproof plan.

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  • But his cry came an instant too late as Shipton plummeted past him, his ice ax swinging in a rip across Dean's calf as he plummeted backward into space, and down to the rocks and churning river below.

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  • The leg wound from Shipton's flailing ice ax had been an eight-stitcher of no permanent consequence but the clump of frozen mountain Dean caught on the head kept him fuzzy and blurred his vision for a day and a half, necessitating the stay.

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  • His muscular back glistened with perspiration as he swung the ax, expertly splitting a chunk of wood.

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  • He shrugged and offered her the ax.

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  • She gripped the ax handle.

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  • Taking a few steps back she gripped the ax half way down on the handle and slammed it down against the block of wood with a dull whack.

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  • The ax blade went about an inch into the wood.

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  • This time the ax sank about four inches into the wood - in another spot.

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  • After a full minute of tugging and grunting she managed to dislodge the ax from the wood.

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  • Again she swung the ax, and this time it went half way through the log.

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  • She relinquished the ax for the glass of tea.

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  • He hefted the ax to his shoulder and grinned down at her.

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  • He lifted the ax, taking aim at a new block of wood.

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  • A Russian proverb says: "What is written down with a pen cannot be hacked away with an ax."

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  • This 40 x 40 ft rock looks as if God took an ax and split the rock in half.

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  • Another Chetnik sexually assaulted women and killed internees, in some cases using an ax to the head.

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  • He tells them that he has got to have an ax and a hickory stake.

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  • In a burst of fear and rage, I ran at it wielding the ax as hard as I could.

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  • Typing ps ax on Saturday usually gave me a half dozen processes to kill.

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  • Consider the general quadratic equation ax 2 + bx + c = 0 where a 0.

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  • Back to top T ax and Indemnities A key question in assessing the value of a settlement deal concerns the impact of tax.

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  • F-Zero ax machine parts Cheats Successfully complete the game in story mode on the hard difficulty setting to unlock the F-Zero AX machine parts.

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  • ax degrees about the x -axis, tilting the data toward the viewer.

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  • ax car and two other vehicles.

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  • ax arcade machine track in first place and save the data to a memory card.

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  • ax n, say, breed successfully and that bx n 2 die from overcrowding.

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  • ax register.

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  • Square matrices A and B are congruent if there exists a non-singular X such that B = X T ax.

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  • Page 25 No need to do Activity 4. Page 25 Graphs of the form ax + by = c.

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  • To see how many processes there are, type ps ax.

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  • AH was really just the higher byte of register ax; and AL the lower byte.

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  • beheaded by the sword, not a clumsy ax.

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  • When using a billhook, slasher or ax, always make sure there is a clear path to swing the tool.

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  • A silver birch, burnt then chopped at with an ax, still looking a bit like a silver birch.

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  • bonanza for bosses Britain's top businessmen will collect up to £ 1m while companies ax final salary schemes for employees.

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  • conspire admitted to conspiring to commit wilful damage over the ax incident.

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  • However, the red Escort was soon to blot it's copybook by failing Ax with no less than three punctures!

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  • Trees were free fuel for poor cottagers, every one of whom had an ax.

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  • A few tragic accidents have resulted from people not wearing crampons or using an ice ax.

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  • The price does not include any accommodation, food or the hire of ice ax crampons, winter boots or any personal winter clothing.

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  • Steve had read a magazine and been shopping; he was experimenting with an ax and ice dagger combination!

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  • I felt I was reading a diatribe from someone with a personal ax to grind, rather than a scholarly or well researched biography.

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  • Finally, the fourth ax of educational quality and inclusive education development is related with pedagogical differentiation, the main content of this paper.

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  • Course Description Solving linear systems Ax = b or finding the eigenvalues of a matrix may appear to be trivial tasks.

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  • Every moment I expect to see the executioner arrive with his ax.

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  • executioner's ax has fallen for the final time.

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  • Unique service for disabled faces ax A cash crisis is threatening the future of a pioneering city counseling center for the disabled.

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  • It is a small, socketed ax head with a loop underneath it for binding it to a wooden haft.

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  • handle of the ax I was using. ' Amber's story has arrived with perfect timing.

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  • This comes from the Fairy Feller in the foreground, his ax poised on high, ready to chop a giant hazelnut in half.

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  • headman's ax.

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  • headsman's ax.

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  • Do you need an ax to split an infinitive?

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  • jadeite ax found by Master Steven Jacob in back garden of 19 in garden soil.

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  • An ax may b e kept near a chopping block to make kindling.

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  • lumberjack sports, ax throwing and ax racing.

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  • He is the legendary lumberman, who supposedly stood at 42 ax handles tall!

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  • There are less noble weapons that you can wield, such as an executioner's ax or spiked maces.

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  • She'd prefer the ax murderer to Troy Marsden, the weird guy who lived on the property up the track.

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  • The giant ogre was smashing the gate with his huge ax each time iron cracked off sending sparks into the air.

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  • At its heart was a sharp iron ax of the weight of seven pounds twelve ounces.

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  • polished stone ax (Fig.

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  • puff of smoke, Des handed Ralph an ax.

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  • purpose-built premises at St Mary Ax in the City of London.

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  • Another man enters clutching a live rooster, presents the protagonist with an ax, then leaves the room.

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  • Then there are night races and blinding sandstorms to struggle through not forgetting the odd throwing ax or javelin that may come your way.

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  • The axeman was present, his ax freshly sharpened.

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  • socketed ax, was recovered with the aid of a metal detector.

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  • spewing frothy blood from his ax wound.

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  • staggered backward, he dropped the ax and tried to pull the chicken body off his face.

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  • He told the story of the young man who wanted his ax ground.

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  • swing the ax I was blinded by snow and the wind pushed me off balance.

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  • traverse icy slopes using crampons, kicking steps in snow or hacking them out in ice with an ax.

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  • trusty pick ax, ropes and dynamite and go on a glorious hunt for treasure.

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  • The firemen's uniform consisted of a helmet, belt, badge and ax.

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  • upstream passage is Ax Wars Inlet.

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  • wield the ax in order to save the Library?

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  • woodcutter's ax.

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  • woodsman's ax has been left | within reach.

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  • Referred to the centre this becomes Ax e +2Hxy+By 2 +C =o; and if the axes of coordinates be the principal axes of the curve, the equation is further simplified to Ax e - By 2 = C, or if the semi-transverse axis be a, and the semiconjugate b, 'x' 2 /a 2 - y 2 /b 2 = I.

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  • Such curves are given by the equation x 2 - y 2 = ax 4 -1bx 2 y 2 +cy 4 .

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  • - 2 ay2' ax, Ux2 ��� a y n ay.

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  • Taking two of the equations ax + +cz) x"' 1 +...

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  • The number of partitions of a biweight pq into exactly i biparts is given (after Euler) by the coefficient of a, z xPy Q in the expansion of the generating function 1 - ax.

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  • It may be written in the form n n-1 2 ax 1 +bx1 x2 +cx 1 x 2 + ...; or in the form n n n=1 n n-2 2 +(1)bx x2+ ?

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  • Other forms are n-1 n-2 2 ax +nbx x +n(n-i)cx x +..., 1121 2 the binomial coefficients C) being replaced by s!(e), and n 1, n-1 1 n-2 2 ax 1 +l i ox l 'x 2 + L ?cx 1 'x2+..., the special convenience of which will appear later.

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  • As between the original and transformed quantic we have the umbral relations A1 = A1a1 d-A2a2, A2 = /21a1+/22a2, and for a second form B1 =A 1 b 1+ A 2 b 2, B 2 =/21bl +�2b2� The original forms are ax, bi, and we may regard them either as different forms or as equivalent representations of the same form.

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  • From the three equations ax = alxl+ a2x2, b.= blxl+b2x2, cx = clxi+c2x2, we find by eliminating x, and x 2 the relation a x (bc)+b x (ca) +c x (ab) =0.

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  • The �th polar of ax with regard to y is n-� a aye i.e.

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  • 32 ax l ay 2 ax2ay1' which, operating upon any polar, causes it to vanish.

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  • l ax 2 2 ax i l This is called the kth transvectant of f over 4); it may be conveniently denoted by (f, (15)k.

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  • The two forms ax, bx, or of, 0, may be identical; we then have the kth transvectant of a form over itself which may, or may not, vanish identically; and, in the latter case, is a covariant of the single form.

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  • Since, If F = An, 4) = By, 1 = I (Df A4) Of A?) Ab A"'^1Bz 1=, (F, Mn Ax I Ax 2 Axe Ax1) J The First Transvectant Differs But By A Numerical Factor From The Jacobian Or Functional Determinant, Of The Two Forms. We Can Find An Expression For The First Transvectant Of (F, �) 1 Over Another Form Cp. For (M N)(F,4)), =Nf.4Y Mfy.4), And F,4, F 5.4)= (Axby A Y B X) A X B X 1= (Xy)(F,4))1; (F,Ct)1=F5.D' 7,(Xy)(F4)1.

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  • Put M 1 For M, N I For N, And Multiply Through By (Ab); Then { (F, C6) } = (Ab) A X 2A Y B X 1 M N I 2 (Xy), ?) 2, = (A B)Ax 1B X 2B Y L I Multiply By Cp 1 And For Y L, Y2 Write C 2, C1; Then The Right Hand Side Becomes (Ab)(Bc)Am Lbn 2Cp 1 M I C P (F?) 2 M { N2 X, Of Which The First Term, Writing C P =, ,T, Is Mn 2 A B (Ab)(Bc)Axcx 1 M 2 N 2 P 2 2222 2 2 _2 A X B X C (Bc) A C Bx M N 2 2 2 M2°N 2 N 2 M 2 2 A X (Bc) B C P C P (Ab) A B B(Ac) Ax Cp 2 = 2 (04) 2 1 (F,0) 2.4 (F,Y') 2 �?; And, If (F,4)) 1 = Km " 2, (F??) 1 1 M N S X X X Af A _Af A Ax, Ax Ax Ax1 Observing That And This, On Writing C 2, C 1 For Y 11 Y 21 Becomes (Kc) K X 'T 3C X 1= (F,0 1 ', G 1; �'�1(F,O) 1 M 1=1 M 2 0`,4)) 2 0, T (Fm 2.4 (0,0 2 .F ' And Thence It Appears That The First Transvectant Of (F, (P) 1 Over 4) Is Always Expressible By Means Of Forms Of Lower Degree In The Coefficients Wherever Each Of The Forms F, 0, 4, Is Of Higher Degree Than The First In X 1, X2.

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  • It is (f = (ab) 2 a n-2 r7 2 =Hx - =H; unsymbolically bolically it is a numerical multiple of the determinant a2 f a2f (32 f) 2� It is also the first transvectant of the differxi ax axa x 2 ential coefficients of the form with regard to the variables, viz.

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  • 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 " �� ' 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.

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  • First observe that with f x =a: = b z = ���,f1 = a l a z ', f 2 = a 2 az-', f x =f,x i +f 2 x i, we find (ab) - (a f) bx - (b f) ax.

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  • Moreover the second term on the left contains (a f)' c -2b z 2 = 2 (a f) k-2b x 2 - (b) /0-2a 2 � if k be uneven, and (af)?'bx (i f) of) '-la if k be even; in either case the factor (af) bx - (bf) ax = (ab) f, and therefore (n-k),bk+1 +M�f = k(n-2)f.(uf)uxn-2k-1; and 4 ' +1 is seen to be of the form f .14+1.

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  • y1 = x 15+f2n; f� y2 =x2-f?n, f .a b = ax+ (a f) n, l; n u 2 " 2 22 2 +` n) u3 n-3n3+...+U 2jn� 3 n Now a covariant of ax =f is obtained from the similar covariant of ab by writing therein x i, x 2, for yl, y2, and, since y?, Y2 have been linearly transformed to and n, it is merely necessary to form the covariants in respect of the form (u1E+u2n) n, and then division, by the proper power of f, gives the covariant in question as a function of f, u0 = I, u2, u3,...un.

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  • The Binary Quadratic.-The complete system consists of the form itself, ax, and the discriminant, which is the second transvectant of the form upon itself, viz.: (f, f') 2 = (ab) 2; or, in real coefficients, 2(a 0 a 2 a 2 1).

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  • =y la xl -i-y2a x2 must also vanish for the root a, and thence ax, and a must also vanish for the same root; which proves that a is a double root of f, and f therefore a perfect square.

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  • If a, a be the linear forms, above defined, he raises the identity ax(0) =ax(aJ3) - (3x(aa) to the fifth power (and in general to the power n) obtaining (aa) 5 f = (a13) 5 az - 5 (a0) 4 (aa) ax?3 -F...

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

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  • Hence, from the identity ax (pq) = px (aq) -qx (ap), we obtain (pet' = (aq) 5px - 5 (ap) (aq) 4 pxg x - (ap) 5 gi, the required canonical form.

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  • The 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.

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  • The complete covariant and contravariant system includes no fewer than 34 forms; from its complexity it is desirable to consider the cubic in a simple canonical form; that chosen by Cayley was ax 3 +by 3 + cz 3 + 6dxyz (Amer.

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  • 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.

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  • Thus what have been called seminvariants are not all of them invariants for the general substitution, but are invariants for the particular substitution xl = X11 + J-s12, X 2 = 112 Again, in plane geometry, the most general equations of substitution which change from old axes inclined at w to new axes inclined at w' =13 - a, and inclined at angles a, l3 to the old axis of x, without change of origin, are x-sin(wa)X+sin(w -/3)Y sin w sin ' _sin ax y sin w a transformation of modulus sin w' sin w' The theory of invariants originated in the discussion, by George Boole, of this system so important in geometry.

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  • then of course (AB) = (ab) the fundamental fact which appertains to the theory of the general linear substitution; now here we have additional and equally fundamental facts; for since A i = Xa i +,ia2, A2= - �ay + X a2, AA =A?-}-A2= (X2 +M 2)(a i+ a z) =aa; A B =AjBi+A2B2= (X2 +, U2)(albi+a2b2) =ab; (XA) = X i A2 - X2 Ai = (Ax i + /-Lx2) (- /-jai + Xa2) - (- / J.x i '+' Axe) (X a i +%Ga^2) = (X2 +, u 2) (x a - = showing that, in the present theory, a a, a b, and (xa) possess the invariant property.

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  • (ab), aa, ab, (xa), ax, xx.

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  • To assist us in handling the symbolic products we have not only the identity (ab) cx + (bc) a x + (ca) bx =0, but also (ab) x x+ (b x) a + (ax) b x = 0, (ab)a+(bc)a s +(ca)a b = 0, and many others which may be derived from these in the manner which will be familiar to students of the works of Aronhold, Clebsch and Gordan.

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  • For the quadratic aoxi +2a i x i x 2 +a 2 x, we have (i.) ax = 7/1x1+2aixix2-I-7/24, (ii.) xx=xi+xzi (ab) 2 =2(aoa2 - ai), a a = a o+712, _ (v.) (xa)ax= i'?- (a2 - ao)xix2 - aix2.

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  • There is no linear covariant, since it is impossible to form a symbolic product which will contain x once and at the same time appertain to a quadratic. (v.) is the Jacobian; geometrically it denotes the bisectors of the angles between the lines ax, or, as we may say, the common harmonic conjugates of the lines and the lines x x .

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  • The linear invariant a s is such that, when equated to zero, it determines the lines ax as harmonically conjugate to the lines xx; or, in other words, it is the condition that may denote lines at right angles.

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  • 3) by F, and its components parallel to the co-ordinate axes by X and Y, we have X= - ax = M(3 cos' 0 - I), Y= - y = M (3 sin 0 cos 0.

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  • Thus, to solve the equation ax e +bx+c = o, we consider, not merely the value of x for which ax 2 +bx+c is o, but the value of ax e +bx+c for every possible value of x.

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  • In this they were completely successful, for they obtained general solutions for the equations ax by = c, xy = ax+by+c (since rediscovered by Leonhard Euler) and cy 2 = ax e + b.

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  • A particular case of the last equation, namely, y 2 = ax e + 1, sorely taxed the resources of modern algebraists.

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  • Denoting them by x, y, so that AB is axis of y and a perpendicular through A the axis of x, and rationalizing (26), we have 2 ax 2 - V 2 Xy 2 - V 2 aAy = o, which represents a hyperbola with vertices at 0 and A.

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  • Ignoring temperature effect, and taking the density as a function of the pressure, surfaces of equal pressure are also of equal density, and the fluid is stratified by surfaces orthogonal to the lines of force; n ap, dy, P d z, or X, Y, Z (4) are the partial differential coefficients of some function P, =fdplp, of x, y, z; so that X, Y, Z must be the partial differential coefficients of a potential -V, such that the force in any direction is the downward gradient of V; and then dP dV (5) ax + Tr=0, or P+V =constant, in which P may be called the hydrostatic head and V the head of potential.

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  • z) = I (ax - I d ?

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  • (2) If the actual motion at any instant is supposed to be generated instantaneously from rest by the application of pressure impulse over the surface, or suddenly reduced to rest again, then, since no natural forces can act impulsively throughout the liquid, the pressure impulse W satisfies the equations I do = I d i dos - ax -u, - - y = -v, Pdz = -t, a =p4)-}-a constant, (4) and the constant may be ignored; and Green's transformation of the energy T amounts to the theorem that the work done by an impulse is the product of the impulse and average velocity, or half the velocity from rest.

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  • u, du +v, du +w, du =o,...,..., (9) ax '?

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  • Uniplanar motion alone is so far amenable to analysis; the velocity function 4 and stream function 1G are given as conjugate functions of the coordinates x, y by w=f(z), where z= x +yi, w=4-Plg, and then dw dod,y az = dx + i ax - -u+vi; so that, with u = q cos B, v = q sin B, the function - Q dw u_vi=g22(u-}-vi) = Q(cos 8+i sin 8), gives f' as a vector representing the reciprocal of the velocity in direction and magnitude, in terms of some standard velocity Q.

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  • Thus if T is expressed as a quadratic function of U, V, W, P, Q, R, the components of momentum corresponding are dT dT dT (I) = dU + x2=dV, x3 =dW, dT dT dT Yi dp' dQ' y3=dR; but when it is expressed as a quadratic function of xi, 'x2, x3, yi, Y2, Y3, U = d, V= dx, ' w= ax dT Q_ dT dT dy 1 dy2 dy The second system of expression was chosen by Clebsch and adopted by Halphen in his Fonctions elliptiques; and thence the dynamical equations follow X = dt x2 dy +x3 d Y = ..., Z ..., (3) = dt1 -y2?y - '2dx3+x3 ' M =..

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  • The Diophantine analysis was a favourite subject with Pell; he lectured on it at Amsterdam; and he is now best remembered for the indeterminate equation ax 2 +1 = y 2, which is known by his name.

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  • To obtain (i) and (ii) together, we show that the volume of a sphere is proportional to the volume of the cube whose edge is the diameter; denoting the constant ratio by aX, the volume of the sphere is Xa 3, and thence, by taking two concentric spheres (cf.

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  • (b) - 49(a), where 4)(x) is any function of x, by [c P(x)]; the area of the trapezette whose bounding ordinates are uo and u m may then be denoted by [Ax.

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  • - 2 To show that the area of a cross-section of a - prismoid is of the form ax e -{- bx -{- c, where x is the distance of the section from one end, we may proceed as in § 27.

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  • Rods of different materials may be used as sounders in a Kundt's dust tube, and their Young's moduli may be compared, since: length of rod Then dO U = - ax dx or dt = - UK.

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  • Now µx 2 is very small compared with Ax, so that x is nearly equal to F/X, and as an approximation, F=Ax+µF 2 /A 2, or x=F/A - µF 2 /A 3.

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  • If the loads are moved a distance Ax to the right, the bending moment becomes M+OM =W I (x+Ax)(l - m)/l+W 2 m{1 - (x+Ox +a) /l} Om = WIDx(l - m)/1 - W20xm/l, and this is positive or the bending moment increases, if W I (l - m) >W 2 m, or if WI/m>W2/(l - m).

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  • 67 b, superimposed and divided so that the length AX represents the load AX, the length AB the load AB, the length YX the reaction YX, and so forth.

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  • To prove this let AB, AB' be the tangents from any point on the line AX.

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  • Similarly AB/ 2 = AX 2 - DX 2 +DP' 2.

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  • aa, ae, af, ag, ah, ak, al, am, an, ap, aq, ar, as, at, au, av, aw, ax, ay, az, bb, In Anodonta these pallial tentacles are confined to a small area surrounding the inferior siphonal notch (fig.

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  • I (6), ax) are the typical pedal ganglia; they are joined to the cerebropleural ganglia by connectives.

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  • Warm mineral springs of note are found at Ax, Aulus and Ussat.

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  • aex; a word common, in different forms, in the Teutonic languages, and akin to the Greek d Lvrt; the New English Dictionary prefers the spelling "ax"), a tool or weapon, taking various shapes, but, when not compounded with some distinguishing word (e.g.

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  • In the geometry of plane curves, the term parabola is often used to denote the curves given by the general equation a' n x n = ym+n, thus ax= y 2 is the quadratic or Apollonian parabola; a 2 x = y 3 is the cubic parabola, a 3 x = y4 is the biquadratic parabola; semi parabolas have the general equation ax n-1 = yn, thus ax e = y 3 is the semicubical parabola and ax 3 = y 4 the semibiquadratic parabola.

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  • Denoting these limits by Pi, Pa we have P1/W=L1H/HK=sin (aX)/cos (0+X),

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  • It appears, moreover, that if 0 be varied P will be least when L1H is at right angles to KLi, in which case Pi =W sin (aX), corresponding to 0 = X.

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  • If a, b, c be the semi-axes of the Binets ellipsoid of G, the quadratic moment with respect to the plane Xx + ~iy + vz =0 will be M(aX + bu + c2vi), and that with respect to a parallel plane ?.x+uy+vz=P (29)

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  • An important property of the diagram is that if points X and x are taken dividing the link CB and the whole acceleration of B about C, namely, cb in the same ratio, then Ax represents the acceleration of the point X in magnitude and direction; cb is called the acceleration image of the rod.

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  • Timid by nature, aware of his impending doom, and ax times justly dissatisfied with himself, he tries all means of reconciling himself to the idea of suicide.

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  • In 1844 Minas published a work, avowedly from a MS. with the superscription Galenus, entitled FaXrlvo13 Eiaa'yw yi, &aX€Krucii.

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  • 1 O rc Stoke 52 t a5 5 Castle ax?

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  • The only roads are bridle-paths, and one municipal road by the Balira valley, connecting Andorra with the high road to Seo de Urgel and Manresa; but in 1904 France and Spain agreed to build a railway from Ax to Ripoll, which would greatly facilitate traffic.

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  • the figure the cortex has been cut away In the order ALCYONto show the axis, ax, and the longiACEA the colony consists tudinal canals, lc, surrounding it.

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  • m, Mouth; mf, mesenterial filament; ax, axis.

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  • a curve is of the first order, second order, third order, &c., according as it is represented by an equation of the first order, ax+by+c = o, or say (1 x, y, 1) = o; or by an equation of the second order, ax 2 +2hxy+by e +2fy+2gx+c=o, say (*I x, y, 1) 2 =o; or by an equation of the third order, &c.; or what is the same thing, according as the equation is linear, quadric, cubic, &c.

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  • Ax, The axis.

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  • Ax' to Ax el, the four segments of the axis with muscular bands.

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  • Considering the equations ax +by +cz =d, a'x +b'y +c' z =d', a"x+b"y+cnz=d" and proceeding to solve them by the so-called method of cross multiplication, we multiply the equations by factors selected in such a manner that upon adding the results the whole coefficient of y becomes = o, and the whole coefficient of z becomes = o; the factors in question are b'c" - b"c', b"c - be", bc' - b'c (values which, as at once seen, have the desired property); we thus obtain an equation which contains on the left-hand side only a multiple of x, and on the right-hand side a constant term; the coefficient of x has the value a(b'c" - b"c') +a'(b"c - bc") +a'(bc' - b'c), and this function, represented in the form a, b,c, a' b'c', a" b" c" is said to be a determinant; or, the number of elements being 32, it is called a determinant of the third order.

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  • Haxey), is hybrid, Ax being the Celtic uisg, water; ey the Anglo-Saxon for island; and holm the Norse word with the same signification.

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  • In 1907 the sanctuary of Athena "of the Brazen House" (X aX KlocKos) was located on the Acropolis immediately above the theatre, and though the actual temple is almost completely destroyed, fragments of the capitals show that it was Doric in style, and the site has produced the longest extant archaic inscription of Laconia, numerous bronze nails and plates and a considerable number of votive offerings, some of them of great interest.

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  • line K'X' parallel to KX such that AX = A'X', then the same curve will be described if we regard K'X' and S' as the given directrix and focus, the eccentricity remaining the same.

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  • In analytical geometry, r the equation axe+2hxy+bye+2gx+2fy+ c = o represents an ellipse when ab > h 2; if the centre of the curve be the origin, the equation is a 1 x 2 +2h 1 xy+b i y 2 =C 1, and if in addition a pair of conjugate diameters are the axes, the equation is further simplified to Ax e +By 2 = C. The simplest form is x 2 /a 2 +y 2 /b 2 = 1, in which the centre is the origin and the major and minor axes the axes of co-ordinates.

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  • Delcass, the French minister for foreign affairs, and on the I 8th of August r904 a convention was signed providing for the construction of (1) the Huesca-Oloron line, (2) a line from Ax les Thermes in the Arige to Ripoll in Catalonia, (3) a line from St Girons in the Arige to Sort, and thence to Lrida.

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  • AX - LES - THERMES, a watering place of south-western France, in the department of Ariege, at the confluence of the Ariege with three tributaries, 26 m.

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  • Ax (Aquae), situated at a height of 2300 ft., is well known for its warm sulphur springs (77°-172° F.), of which there are about sixty.

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  • Diagrammatic longitudinal section of cone, showing the axis (ax) bearing the bracts (br) with peltate sporangiophores (sp) springing from their axils; sm, sporangia.

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  • Part of cone, showing the axis (ax) bearing peltate sporangiophores (sp) without bracts; sm, sporangia.

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  • Diagrammatic longitudinal section of the cone, showing the axis (ax) bearing alternate whorls of bracts (br) and peltate sporangiophores (sp) with their sporangia (sm).

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  • cy, Stele of axis (Ax).

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  • The dorsal segments are sterile, corresponding to the bracts of Sphenophyllum Dawsoni, while the ventral segments constitute peltate sporangiophores, each bearing four sporangia, just as in a ax FIG.

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  • ax, Axis.

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  • angia, usually of very large ax, Axis, bearing the sporophylls (sph), size compared with those of on each of which a sporangium most recent Lycopods, have (sm) is seated.

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  • One carried a gun, one had a pitchfork, and the third had an ax.

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  • He was armed with a musketoon (which he carried rather as a joke), a pike and an ax, which latter he used as a wolf uses its teeth, with equal ease picking fleas out of its fur or crunching thick bones.

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  • Tikhon with equal accuracy would split logs with blows at arm's length, or holding the head of the ax would cut thin little pegs or carve spoons.

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  • He had a musketoon over his shoulder and an ax stuck in his girdle.

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  • So I went for them with my ax, this way: 'What are you up to?' says I.

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  • Another man enters clutching a live rooster, presents the protagonist with an ax, then leaves the room.

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  • Then there are night races and blinding sandstorms to struggle through not forgetting the odd throwing ax or javelin that may come your way.

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  • The axeman was present, his ax freshly sharpened.

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  • A single fragment of bronze, probably part of a socketed ax, was recovered with the aid of a metal detector.

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  • The final nightmare is missing a nice close-up of the father 's head spewing frothy blood from his ax wound.

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  • Instead, if you are hit by an ax, or are shot, you get a blood splatter across the screen.

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  • As Beaker staggered backward, he dropped the ax and tried to pull the chicken body off his face.

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  • He told the story of the young man who wanted his ax ground.

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  • Every time I wanted to swing the ax I was blinded by snow and the wind pushed me off balance.

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  • Then we learned to traverse icy slopes using crampons, kicking steps in snow or hacking them out in ice with an ax.

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  • Use your trusty pick ax, ropes and dynamite and go on a glorious hunt for treasure.

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  • The firemen 's uniform consisted of a helmet, belt, badge and ax.

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  • On the left, the upstream passage is Ax Wars Inlet.

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  • He carries a working flashlight in his hand and a fireman 's ax made from padded velour fabric is fixed to his belt.

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  • Where would you wield the ax in order to save the Library?

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  • Suddenly, they heard the clean chopping strokes of the woodcutter 's ax.

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  • An old, woodsman 's ax has been left | within reach.

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  • He finally quit after smoking for forty-five years, although he said in the first few days after quitting he "felt like an ax murderer."

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  • Myrica Cerifera - ax Myrtle) and also M.

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  • Jazz Guitar: A Belgian site, Jazz Guitar is for the guitar player who prefers to get a smoother sound out of her ax.

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  • Lizzie Borden House: This house is the site of the famous Lizzie Borden ax murders made famous in the children's nursery rhyme.

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  • Many shows felt the ax in the 2008-2009 television season, due in no small part to financial reasons.

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  • Among them are The Birdman of Alcatraz with Burt Lancaster, Escape from Alcatraz with Clint Eastwood, The Rock, and So I Married an Ax Murderer.

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  • But his cry came an instant too late as Shipton plummeted past him, his ice ax swinging in a rip across Dean's calf as he plummeted backward into space, and down to the rocks and churning river below.

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  • She gripped the ax handle.

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  • Taking a few steps back she gripped the ax half way down on the handle and slammed it down against the block of wood with a dull whack.

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  • The ax blade went about an inch into the wood.

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  • From the three equations ax = alxl+ a2x2, b.= blxl+b2x2, cx = clxi+c2x2, we find by eliminating x, and x 2 the relation a x (bc)+b x (ca) +c x (ab) =0.

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  • 32 ax l ay 2 ax2ay1' which, operating upon any polar, causes it to vanish.

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  • The process of transvection is connected with the operations 12; for?k (a m b n) = (ab)kam-kbn-k, (x y x y or S 2 k (a x by) x = 4))k; so also is the polar process, for since f k m-k k k n - k k y = a x by, 4)y = bx by, if we take the k th transvectant of f i x; over 4 k, regarding y,, y 2 as the variables, (f k, 4)y) k (ab) ka x -kb k (f, 15)k; or the k th transvectant of the k th polars, in regard to y, is equal to the kth transvectant of the forms. Moreover, the kth transvectant (ab) k a m-k b: -k is derivable from the kth polar of ax, viz.

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  • Since, If F = An, 4) = By, 1 = I (Df A4) Of A?) Ab A"'^1Bz 1=, (F, Mn Ax I Ax 2 Axe Ax1) J The First Transvectant Differs But By A Numerical Factor From The Jacobian Or Functional Determinant, Of The Two Forms. We Can Find An Expression For The First Transvectant Of (F, �) 1 Over Another Form Cp. For (M N)(F,4)), =Nf.4Y Mfy.4), And F,4, F 5.4)= (Axby A Y B X) A X B X 1= (Xy)(F,4))1; (F,Ct)1=F5.D' 7,(Xy)(F4)1.

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  • It is (f = (ab) 2 a n-2 r7 2 =Hx - =H; unsymbolically bolically it is a numerical multiple of the determinant a2 f a2f (32 f) 2� It is also the first transvectant of the differxi ax axa x 2 ential coefficients of the form with regard to the variables, viz.

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  • 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 " �� ' 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.

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  • If the forms be ax, b2, cy,...

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  • Moreover the second term on the left contains (a f)' c -2b z 2 = 2 (a f) k-2b x 2 - (b) /0-2a 2 � if k be uneven, and (af)?'bx (i f) of) '-la if k be even; in either case the factor (af) bx - (bf) ax = (ab) f, and therefore (n-k),bk+1 +M�f = k(n-2)f.(uf)uxn-2k-1; and 4 ' +1 is seen to be of the form f .14+1.

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  • y1 = x 15+f2n; f� y2 =x2-f?n, f .a b = ax+ (a f) n, l; n u 2 " 2 22 2 +` n) u3 n-3n3+...+U 2jn� 3 n Now a covariant of ax =f is obtained from the similar covariant of ab by writing therein x i, x 2, for yl, y2, and, since y?, Y2 have been linearly transformed to and n, it is merely necessary to form the covariants in respect of the form (u1E+u2n) n, and then division, by the proper power of f, gives the covariant in question as a function of f, u0 = I, u2, u3,...un.

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  • The process of transvection is connected with the operations 12; for?k (a m b n) = (ab)kam-kbn-k, (x y x y or S 2 k (a x by) x = 4))k; so also is the polar process, for since f k m-k k k n - k k y = a x by, 4)y = bx by, if we take the k th transvectant of f i x; over 4 k, regarding y,, y 2 as the variables, (f k, 4)y) k (ab) ka x -kb k (f, 15)k; or the k th transvectant of the k th polars, in regard to y, is equal to the kth transvectant of the forms. Moreover, the kth transvectant (ab) k a m-k b: -k is derivable from the kth polar of ax, viz.

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  • If the forms be ax, b2, cy,...

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  • An ax will be useful, a hunting spear not bad, but a three-pronged fork will be best of all: a Frenchman is no heavier than a sheaf of rye.

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