W Sentence Examples

In Phoenician itself and in the other Semitic alphabets the position of the middle legs of the W is altered so that the symbol takes such forms as or V or w, ultimately ending sometimes in a form like K laid sideways, he.

Subsequently he became, w ith Benjamin Franklin and Arthur Lee, one of the regularly accredited commissioners to France from Congress.

Call this class w; then to say that x is a w is equivalent to saying that x is not an x.

Accordingly, to say that w is a w is equivalent to saying that w is not a w.

He was succeeded by his brother Osman, whose three years' reign w as marked by no political event of special 1754-1757.

Then if j, J be the original and transformed forms of an invariant J= (a1)wj, w being the weight of the invariant.

Let this number be denoted by (w; 0, n).

In order to obtain the seminvari ants we would write down the (w; 0, n) terms each associated with a literal coefficient; if we now operate with 52 we obtain a linear function of (w - I; 8, n) products, for the vanishing of which the literal coefficients must satisfy (w-I; 0, n) linear equations; hence (w; 8, n)-(w-I; 0, n) of these coefficients may be assumed arbitrarily, and the number of linearly independent solutions of 52=o, of the given degree and weight, is precisely (w; 8, n) - (w - I; 0, n).

It is shown in the article on Combinatorial Analysis that (w; 0,n) is the coefficient of a e z w in the ascending expansion of the fraction 1-a.

We may, by a well-known theorem, write the result as a coefficient of z w in the expansion of 1 - z n+1.

Putting n equal to co, in a generating function obtained above, we find that the function, which enumerates the asyzvgetic seminvariants of degree 0, is 1 1-z2.1-z3.1-z4....1-z0 that is to say, of the weight w, we have one form corresponding to each non-unitary partition of w into the parts 2, 3, 4,...0.

T Aw 6(23)B A Series Of 2(W 2) Or Of 2(W I) Forms According As W Is Even Or Uneven.

Their Number For Any Weight W Is The Number Of Ways Of Composing W 3 With The Parts I, 2, And Thus The Generating Function Is Verified.

For w = i the form is A i ai+Bib i, which we may write aob l -albo = ao(I) b -(I)abo; the remaining perpetuants, enumerated by z I - 2' have been set forth above.

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.

Of the quadratic axe+2bxy+cy2, he discovered the two invariants ac-b 2, a-2b cos w+c, and it may be verified that, if the transformed of the quadratic be AX2=2BXY+CY2, sin w 2 AC -B 2 =) (ac-b2), sin w A-2B cos w'+C = (sin w'1 2(a - 2bcosw+c).

The potential at any point due to a magnetic shell is the product of its strength into the solid angle w subtended by its edge at the given point, or V = Fu.

Forces acting on a Small Body in the Magnetic Field.-If a small magnet of length ds and pole-strength m is brought into a magnetic field such that the values of the magnetic potential at the negative and positive poles respectively are V 1 and the work done upon the magnet, and therefore its potential energy, will be W =m(V2-Vi) =mdV, which may be written W =m d s- = M d v= - MHo = - vIHo, ds ds where M is the moment of the magnet, v the volume, I the magnetization, and Ho the magnetic force along ds.

Denoting by W the work in ergs done upon a cubic centimetre of the metal (=_fHdB or f HdI), he finds W =nips approximately, where n 47r is a number, called the hysteretic constant, depending upon the metal, and B is the maximum induction.

But though a formula of this type has no physical significance, and cannot be accepted as an equation to the actual curve of W and B, it is, nevertheless, the case that by making the index e =1.6, and assigning a suitable value to r t, a formula may be obtained giving an approximation to the truth which is sufficiently close for the ordinary purposes of electrical engineers, especially when the limiting value of B is neither very great nor very small.

Working with two different specimens, he found that the hysteresis loss in ergs per cubic centimetre (W) was fairly represented by o 00125B 1 6 and o o0101B 1 ' 6 respectively, the maximum induction ranging from about 300 to 3000.

After pointing out that, since the magnetization of the metal is the quantity really concerned, W is more appropriately expressed in terms of I, the magnetic moment per unit of volume, than of B, he suggests an experiment to determine whether the mechanical work required to effect the complete magnetic reversal i Phil.

He made use of the expression F =Wg=27r12+HI, where W is the weight in grammes per square centimetre of sectional area, and g is the intensity of gravity which was taken as.

W is a weight capable of sliding from end to end of the yoke along a graduated scale.

The weight W is moved along the scale until the yoke just tilts over upon the stop S; the distance of W from its zero position is then, as can easily be shown, proportional to F, and therefore to B 2, and approximately to I 2.

If the field is uniform, H=O/wd, where 0 is the rotation, d the thickness of the substance arranged as a plate at right angles to the direction of the field, and w Verdet's constant for the substance.

In the case of iron containing 7.5% of tungsten (W), the residual induction had a remarkably high value; the coercive force, however, was not very great.

There was considerable hysteresis, the energy-loss per cycle being fairly represented by W =0.000549513 2 ' 238.

Annexed are values of Io 6 K for the different salts examined, w being the weight of the salt per c.c. of the solution.

Thus it occurs in a magical book of Moses, w hich has been edited from a Leiden papyrus of the 3rd or 4th century by Dieterich (Abraxas, 109).

As a centre from which good natural roads lead N.,N.E.,W.

We cannot solve the equation 7X =4s.; but we are accustomed to subdivision of units, and we can therefore give a meaning to X by inventing a unit w s.

The frontier w ith Hungary was the last to be regulated.

We conclude that, with a grating composed of transparent and opaque parts, the utmost light obtainable in any one spectrum is in the first, and there amounts to I/wr 2, or about 6, and that for this purpose W a and d must be equal.

A similar expression can be found for Q'P - Q"A; and thus, if Q' A =v, Q' AO = where v =a cos (0", we get - - -AQ' = a sin w (sin 4 -sink") - - 8a sin 4 w(sin cktan 4 + sin 'tan cl)').

If w now relate to the edge of the grating, on which there are altogether n lines, no- = 2a sin w, and the value of the last term in (I o) becomes no- sin 3w sin O'tan 0', - 1 1 - 6 mnX sin' w tan 0'.

This expresses the retardation of the extreme relatively to the central ray, and is to be reckoned positive, whatever may be the signs of w, and 0 .

If the semi-angular aperture (w) be T 36, and tan 0' might be as great as four millions before the error of phase would reach 4X.

If w be the apparent magnitude of the source seen from A, wb should be much smaller than the above quantity, or w

The beaches which had been selected were, enumerating from right to left, " S " in Morto Bay, " V " and " W " on either side of Cape Helles at the south-western end, and " X " and " Y " on the outer shore; " V " and " W " were regarded as of primary importance, as those two beaches offered suitable landing places from the point of view of subsequent operations.

The attacks at " S " and " Y " were intended to be subsidiary; but great importance was attached to that at " X " owing to the vicinity of this point to " W."

As it turned out, the actual disembarkations at " S," " X " and " Y " were carried out without any very great difficulty; but the troops detailed for " W " beach only gained a footing after incurring very heavy losses and by a display of indomitable resolution, while at " V " the operation went very near to failing altogether.

But the forces which had landed at " W " and " X " beaches had joined hands, the one battalion detailed for " S " beach had secured a good position, and during the night the troops still left aboard the " River Clyde " contrived to disembark.

What was left of the force originally detailed for the landing at " V " beach contrived during the early hours by stern fighting to occupy some high ground hard by, and also to join hands with the troops landed at " W " beach.

Additional infantry was got ashore at " W " and " X " beaches, the first elements of the French division began disembarking at " V " beach in the afternoon, and before evening touch had been gained with the battalion that had made good at " S " beach.

Ottoman guns dominated the entire territory which the invaders had succeeded in the course of two months in conquering, as well as " V " and " W " beaches which were the landing-places chiefly used by them.

Embarkation operations were carried on almost entirely at " V " and " W " beaches, at both of which there were provisional breakwaters in existence furnishing some shelter when there was an onshore breeze.

The right half of the British were to withdraw by " V " beach and the left half by " W " beach, except that the final detachments on the extreme left, representing the 13th Division, were to be got off at Gully beach.

But the detachments designated for Gully beach could not all be got off at the exposed point, and those left over had to march on to " W " beach at the last moment and were not afloat till nearly 4 A.M., their embarkation being effected with great difficulty owing to the surf.

It is used to determine the density of a body experimentally; for if W is the weight of a body weighed in a balance in air (strictly in vacuo), and if W' is the weight required to balance when the body is suspended in water, then the upward thrust of the liquid (I) (2) "F r an Minim ' 'i n or weight of liquid displaced is W-W, so that the specific gravity (S.G.), defined as the ratio of the weight of a body to the weight of an equal volume of water, is W/(W-W').

Putting (12) a vortex line is defined to be such that the tangent is in the direction of w, the resultant of, n, called the components of molecular rotation.

Over a concentric cylinder, external or internal, of radius r=b, 4,'=4,+ Uly =[U I - + Ui]y, (4) and 4" is zero if U 1 /U = (a 2 - b2)/b 2; (5) so that the cylinder may swim for an instant in the liquid without distortion, with this velocity Ui; and w in (I) will give the liquid motion in the interspace between the fixed cylinder r =a and the concentric cylinder r=b, moving with velocity U1.

If the liquid is reduced to rest at infinity by the superposition of an opposite stream given by w = - Uz, we are left with w = Ua2/z, (6) =U(a 2 /r) cos 0= Ua2x/(x2+y2), (7) 4, = -U(a 2 /r) sin 0= -Ua2y/( x2+y2), (8) giving the motion due to the passage of the cylinder r=a with velocity U through the origin 0 in the direction Ox.

The expression for w in (i) § 29 may be increased by the addition of the term im log z =-m0 + im log r, (1) representing vortex motion circulating round the annulus of liquid.

Consider the streaming motion given by w =m =a+si, (5) 4=m ch (n -a)cos(-0), p=m sh(n-a)sin(-13).

In a similar way the more general state of motion may be analysed, given by w =r ch2('-y), y =a+, i, (26) as giving a homogeneous strain velocity to the confocal system; to which may be added a circulation, represented by an additional term in w.

The polar equation of the cross-section being rI cos 19 =al, or r + x = 2a, (3) the conditions are satisfied by = Ur sin g -2Uairi sin IB = 2Uri sin 10(14 cos 18a'), (4) 1J/ =2Uairi sin IO = -U1/ [2a(r-x)], (5) w =-2Uaiz1, (6) and the resistance of the liquid is 2lrpaV2/2g.

Denoting the cross-section a of a filament by dS and its mass by dm, the quantity wdS/dm is called the vorticity; this is the same at all points of a filament, and it does not change during the motion; and the vorticity is given by w cos edS/dm, if dS is the oblique section of which the normal makes an angle e with the filament, while the aggregate vorticity of a mass M inside a surface S is M - l fw cos edS.

The components of velocity of the moving origin are denoted by U, V, W, and the components of angular velocity of the frame of reference by P, Q, R; and then if u, v, w denote the components of fluid velocity in space, and u', v', w' the components relative to the axes at a point (x, y, z) fixed to the frame of reference, we have u =U +u' - yR +zQ, v =V +v -zP +xR, w=W +w -xQ +yP.

As an application of moving axes, consider the motion of liquid filling the ellipsoidal case 2 y 2 z2 Ti + b1 +- 2 = I; (1) and first suppose the liquid be frozen, and the ellipsoid l3 (4) (I) (6) (9) (I o) (II) (12) (14) = 2 U ¢ 2, (15) rotating about the centre with components of angular velocity, 7 7, f'; then u= - y i +z'i, v = w = -x7 7 +y (2) Now suppose the liquid to be melted, and additional components of angular velocity S21, 522, S23 communicated to the ellipsoidal case; the additional velocity communicated to the liquid will be due to a velocity-function 2224_ - S2 b c 6 a 5 x b2xy, as may be verified by considering one term at a time.

When the liquid is bounded externally by the fixed ellipsoid A = A I, a slight extension will give the velocity function 4 of the liquid in the interspace as the ellipsoid A=o is passing with velocity U through the confocal position; 4 must now take the formx(1'+N), and will satisfy the conditions in the shape CM abcdX ¢ = Ux - Ux a b x 2+X)P Bo+CoB I - C 1 (A 1 abcdX, I a1b1cl - J o (a2+ A)P and any'confocal ellipsoid defined by A, internal or external to A=A 1, may be supposed to swim with the liquid for an instant, without distortion or rotation, with velocity along Ox BA+CA-B 1 -C1 W'.

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 =..

But the presence of the medium makes the effective inertia depend on the direction of motion with respect to the external shape of the body, and on W' the weight of fluid medium displaced.

For the body alone the resultant of the components of momentum W V -cos andW V sin 0 is W V -sec. lb, acting along 00', and so is unaltered.

If p denotes the density of the air or medium W' = sird 3 xp, (23) W' I p __ W I -1 3 k12 I k22 x2 ±i a 2= 101-1 3 '111 2= 2 tan g S = Q (l - a) x 2+ I (26) in which a/p may be replaced by 800 times the S.G.

I +W a W a), ' (k) 4 (I I) I+ w- R For a shot in air the ratio W'/W is so small that the square may be neglected, and formula (II) can be replaced for practical purpose in artillery by tan26= n2 = W i (0 - a) (k ð)7()4, (12) if then we can calculate /3, a, or (3-a for the external shape of the shot, this equation will give the value of 6 and n required for stability of flight in the air.

The cuneiform system of writing Semitic. w as still in process of growth when it was borrowed influence p g and adapted by the new comers, and the Semitic Babylonian language was profoundly influenced by the older language of the country, borrowing its words and even its grammatical usages.

In German, however, V is used with the same value as F, while W takes the value that V has in English.

Early borrowings like wine (Latin vinum), wall (Latin vallum), retain the w sound and are therefore spelt with w.

As an auxiliary to air cooling the stack may be cooled by a slow stream of water trickling down the outside of the pipes, or, in certain cases, cold water may be injected into the condenser in the form of a spray, w here it meets the ascending vapours.

In algebra, if a be a real positive quantity and w a root of unity, then a is the amplitude of the product aw.

Moreover, the three solids S,D and W will differ in minute structure and therefore, probably, in mechanical properties.

Consider the integral W dx dy dz .

At the end of the 6th century the exarchate included Istria; the maritime part of Venetia as distinct from the interior which was in the hands of the Lombard kings at Pavia; the exarchate proper, or territory around Ravenna on the eastern side of the Apennines, to which was added Calabria, which at that period meant the heel and not the toe of the boot; the Pentapolis, or coast from Rimini to Ancona with the interior as far as the mountains; the duchy of Rome, or belt of territory connecting the Pentapolis with the western coast, the coast of Naples, w i th Bruttium the toe of the boot, the modern Calabria, and Liguria, or the Riviera of Genoa.

The remainder is equivalent to the external work, W, done by the body in expanding or otherwise, which can be utilized for mechanical purposes, and ceases to exist as heat in the body.

This equation is generally true for any series of transformations, provided that we regard H and W as representing the algebraic sums of all the quantities of heat supplied to, and of work done by the body, heat taken from the body or work done on the body being reckoned negative in the summation.

The area ABCD, representing the work, W, per cycle, is the difference (H' - H") of the quantities of heat absorbed and rejected at the temperatures 0 and 0".

As the temperature 0" is lowered, the area of the cycle increases, but since W can never exceed H', there must be a zero limit of temperature at which the pressure would vanish and the area of the cycle become equal to the whole heat absorbed at the higher temperature.

The change of the intrinsic energy in passing from one state to another, as from B to C is represented by the addition of the heat-area H= Bczz', and the subtraction of the work-area W = BCcb.

The same result follows if the work W =RT log e ?'

Joule's experiments on the equivalence of W and H were not sufficiently precise to decide the question.

This most fundamental point was finally settled by a more delicate test, devised by Lord Kelvin, and carried out in conjunction with Joule (1854), which showed that the fundamental assumption W =H in isothermal expansion was very nearly true for permanent gases, and that F'(t) must therefore vary very nearly as J/T.

Combining this with the first law, for a Carnot cycle of finite range, if H is the heat taken in at 0', and H" is the heat rejected at 0", the work W done in the cycle is equal to the difference H' - H", and we have the simple relations, W/(0' - o") =H'/o' =H" o".

Then by relations (2) the heat, H, absorbed in the isothermal change BC, is to the work, W, done in the cycle ABCD in the ratio of o to (o' - o").

W, External work done by fluid.

In this alphabet the Greek letter p (or rather a very similar letter with the loop a little lower down) is used to represent sh, and there are some peculiarities in the use of o apparently connected with the expression of the sounds h and w.

Calling the weight of the empty vessel w, when filled with the liquid W, and when filled with the standard substance W l, it is obvious that W - w, and W1 - w, are the weights of equal volumes of the liquid and standard, and hence the relative density is (W - w)/(Wi - w).

It is readily seen that W+W i - W 2 is the weight of the liquid displaced by the solid, and therefore is the weight of an equal volume of liquid; hence the relative density is W/(W+Wi - W2).

If W be the weight in air, and W, the weight in water, then W, is always less than W, the difference W - W1 representing the weight of the water displaced, i.e.

Hence W/(W - W,) is the relative density or specific gravity of the body.

The principle is readily adapted to the determination of the relative densities of two liquids, for it is obvious that if W be the weight of a solid body in air, W, and W2 its weights when immersed in the liquids, then W - W, and W - W 2 are the weights of equal volumes of the liquids, and therefore the relative density is the quotient (W - W,)/(W - W2).

If W be the weight of the experimental solid in air, w the weight of the sinker in water, and W, the weight of the solid plus sinker in water, then the relative density is given by W/(W+w - Wi).

There are several corrections of the formula O =W/(W - W1) necessary to the accurate expression of the density.

Less accurate formulae are =p W/(W - W 2), the factor involving the density of the air, and the coefficient of the expansion of the solid being disregarded, and 0 =W/(W - W 1), in which the density of water is taken as unity.

The counterpoise w balances the head about its axis of rotation.

A similar position has also been taken up by Schiirer, 3 Holtzmann,4 Gebhardt, 5 W y llie,' and van Manen.

The heating at the surfaces, the variations in their smoothness, and the variations of the lubrication make continuously variable, and necessitate frequent adjustment of W or of the nuts.

A flexible steel band, lined with wood blocks, is gripped on the motor fly-wheel or pulley by a screw A, which, together with W, is adjusted to hold the brake steady.

The external forces holding the brake from turning are W, distant R from the axis, and the reaction, W 1 say, of the lever against the fixed pin P, distant R1 from the axis.

The moment of W 1 may be positive or negative.

When the wheel is turning in the direction indicated, the forces holding the band still are W, and p, the observed pull on the spring balance.

Both these forces usually act at the same radius R, the distance from the axis to the centre line of the rope, in which case the torque T is (W-p)R, and consequently the brake horse-power is (W - p)RX21rN, When µ 33,000 changes the weight W rises or falls against the action of the spring balance until a stable condition of running is obtained.

The ratio W/p increases very rapidly as 0 is increased,, and therefore, by making 0 sufficiently large, p may conveniently be made a small fraction of W, thereby rendering errors of observation of the spring balance negligible.

Weights W and w are adjusted to the torque.

The number of molecules of the first kind of gas, whose components of velocity lie within the ranges between u and u+du, v and v+dv, w and w+dw, will, by formula (5), be v?l (h 3 m 3 /7 3)e hm (u2+v2+w2)dudvdw (9) per unit volume.

Thus the contribution to the total impulsive pressure exerted on the area dS in time dt from this cause is mu X udtdS X (11 3 m 3 /,r 3)e hm (u2+v2+w2 )dudvdw (I o) The total pressure exerted in bringing the centres of gravity of all the colliding molecules to rest normally to the boundary is obtained by first integrating this expression with respect to u, v, w, the limits being all values for which collisions are possible (namely from - co too for u, and from - oo to + oo for v and w), and then summing for all kinds of molecules in the gas.

The aggregate amount of these pressures is clearly the sum of the momenta, normal to the boundary, of all molecules which have left dS within a time dt, and this will be given by expression (pp), integrated with respect to u from o to and with respect to v and w from - oo to +oo, and then summed for all kinds of molecules in the gas.

The shunt coil Sh and the series coil Se are connected together at one point, and the remaining terminals of the dynamo and shunt coil must be connected to a " good earth," which is generally the gas or water pipes w of the building.

Sometimes the badges took the shape of small ampullae, or vases, as in the case of the badges of the shrine of Our Lady of Walsingham, which were marked with a W and crown.

Let E be the bulk modulus of elasticity, defined as increase of pressure = decrease of volume per unit volume where the pressure increase is so small that this ratio is constant, w the small increase of pressure, and - (dy/dx) the volume decrease, then E=e/(- dy/dx) or w Æ= - dy/dx (I) This gives the relation between pressure excess and displacement.

Let us seek a more'exact value for w.

We have U-Fw= D/T 1 and U - w = D/T .

It makes with it an interval = 1 w 6 - j 2 rather less than a comma; so that the same string in the piano may serve for both.

Let the velocity of the air from S to R be w, and let U be the velocity of sound in still air.

Then the n waves occupy a space U + w - u.

If he were at rest, it would be the waves in length U + w, for the wave passing him at the beginning of a second would be so far distant at the end of the second.

But through his motion v in the second, he receives only the waves in distance U + w - v.

Since there are n waves in distance U + w - u the number he actually receives is n(U + w - v)/(U + w - u).

It is therefore W m = (E 1 U) 27rUa/A = 27rnpUa.

If w is the total pressure excess, and if y is the total displacement at x, then w = E Xchange of volume _original volume = - Edy/dx.

If p is the density at A, and w the cross-section, then the momentum carried past A is pc,(U - u) 2.

The movable bridge in its closed position must be proportioned like a fixed w'?

If w is the weight of a locomotive in tons, r the radius of curvature of the track, v the velocity in feet per sec.; then the horizontal force exerted on the bridge is wv 2 /gr tons.

Let W 1 be the total live load, Wf the total flooring load on a bridge of span 1, both being considered for the present purpose to be uniform per ft.

Since in designing a bridge W1-l-Wt is known, k(W1+Wf) can be found from a provisional design in which the weight W 2 is neglected.

Let w 1, w 2 be the weights of main girders per ft.

Now let w1', w 2 ' be the girder weights per ft.

Then w 3 = (w l +w 2)1 2 /(Cds-1 2) = (w i -Fw 2)lr/ (Cs -lr), where C is a constant for any type of girder.

Hence the formula is more useful in the form w = (w i +w2)1 2 / (Kd -1 2) = (wl +w 2)lr/ (K -lr) where k= (wl+w2-1-w3)lr/w3 is to be deduced from the data of some bridge previously designed with the same working stresses.

That at the left abutment is R 1 =W 1+ W 2+ ...-R2.

Let s be the perpendicular from 0, the join of C and T on the direction of S; t the perpendicular from A, the join of C and S on the direction of T; and c the perpendicular from B, the join of S and T on the direction of C. Taking moments about 0, Rx - W 1 (x+a) - W 2 (x+2a) =Ss; taking moments about A, R3a-W 1 2a-W 2 a =Tt; and taking moments about B, Rea-W I a = Cc.

The reactions at the abutments are R 1 = Wm/1 and R2 = W (l - m)/l.

Next let the girder carry a uniform load w per ft.

The distribution of total shear, due to a dead load w 1 per ft.

The greatest shear at C may occur with W I at C. If W 1 passes FIG.

But W' these are the average A 3 ' 3 C 3 13 B loads per ft.

C A W W 3 j ?..a... ? ?-- -- - n--- --y ?

Let W 1, W2, W3 traverse the girder from the left at fixed distances a, b.

For the position shown the distribution of bending moment due to W 1 is given by ordinates of the triangle 000 A'CB'; that due to W2 by ordin al, W, WW1 W„ ates of A'DB'; and that due to W3 by ordinates of A EB'.

The total moment at WI, due to three loads, is the sum mC-Fmn--Emo of the intercepts which the triangle sides cut off from the vertical under W 1.

Till W 1 has advanced a distance a only one load is on the girder, and the curve A"F gives bending moments due to W 1 only; as W1 advances to a distance a+b, two loads are on the girder, and the curve FG gives moments due to W 1 and W2.

Let w e be the load per ft.

Then w e may be termed the uniform load per ft.

But it is not difficult to find w e, approximately enough for practical purposes, very simply.

From these equations a value of W e can be obtained.

Hence the moment of the load on Am at C is wy0m, and the moment of a uniform load over any portion of the girder is w X the area of the influence curve Ip' G' E ' under that portion.

For a uniformly distributed load w per ft.

Similarly, rs = W (x +a) 11 is the shear on DB.

Thus for a beam supported at the ends and loaded with w per inch length M =w (a 2 - x 2), where a is the half span.

The large loans raised in Europe, the first instalment of which Byron had himself brought over, while providing the Greeks with the sinews of war, provided seco n d civil w them also with fresh material for strife.

We shall make the natural supposition that motion of the aether, say with velocity (u,v,w) at the point (x,y,z), is simply superposed on the velocity V of the optical undulations through that medium, the latter not being intrinsically altered.

But in the transition from molecular theory to the electrodynamics of extended media, all magnetism has to be replaced by a distribution of current; the latter being now specified by volume as well as by flow so that (u,v,w) ST is the current in the element of volume 6T.

Mecherzynski, in his "History of Eloquence in Poland" (Historya wymowy w Polsce), especially praises his two funeral sermons on the burial of Anna Jagiellonka, widow of Stephen Batory, and Anna of Austria, first wife of Sigismund III.

The "Sketch of the History of Poland" (Dzieje Polskie w zarysie) by Michael Bobrzynski, born in 1849 in Cracow (professor of Polish and German law), is a very spirited work, and has given rise to a great deal of controversy on account of the opposition of many of its views to those of the school of Lelewel.

We first determine the time t in seconds required for the velocity of a shot, d inches in diameter and weighing w lb, to fall from any initial velocity V(f/s) to any final velocity v(f/s).

Equating the muzzle-energy and the work in foot-tons (2) E= w V 2 _XM.E.P.

The function log x as x increases from o towards w steadily increases from - co towards +co.

Rao D 0 UEU i-an;, w ma.

The first to take advantage of these altered conditions was William Tyndale, " to whom," as Dr Westcott says,' " it William has been allowed more than to any other man to give W Tyndale.i ts characteristic shape to the English Bible."

The three leading colonial cities, Philadelphia, w York and Boston, grew six-fold in the I 8th century, and fiftyfo in the next.

The head-master's house (W) is opposite, built against the side wall of the church.

Close to the entrance was a chapel, where is now the church of St Olaf (W), in which the new-corners paid their devotions immediately on their arrival.

The small cloister is at W, where were the carols or cells of the scribes, with the library (P) over, reached by a turret staircase.

As peculiarities of arrangement may be noticed the position of the kitchen (Q), between the refectory and calefactory, and of the infirmary (W) (unless there is some error in its designation) above the river to the west, adjoining the guest-houses (XX).

I (I), and, after passing as far forward as the mouth w in fig.

The Above Expression Must Therefore Be Diminished By The Number Of Units In 4, Or By () W (This Notation Being Used To Denote The Quotient, In A Whole Number, That Arises From Dividing X By 4).

Hence In The Julian Calendar The Dominical Letter Is Given By The Equation L= 7M 3 X () W This Equation Gives The Dominical Letter Of Any Year From The Commencement Of The Era To The Reformation.

As 1600 Was A Leap Year, The First Correction Of The Julian Intercalation Took Place In 1700; Hence, Taking C To Denote The Number Of The Century As Before, The Correction Becomes (C 16) (C 416) W, Which.

It May Be Remarked, That As A (C _ 17) W' The Value Of A Will Be O Till C 17C 17 =25 Or C =42; Therefore, Till The Year 4200, A May Be Neglected In The Computation.

Then, by the principle of Archimedes, W = Vwo; or wo = W/V.

Also W = (V +IA)w i; or w1=W/(V+/A), w p =W/(V+plA), and wn =W/(Vd-nIA), or the densities of the several liquids vary inversely as the respective volumes of the instrument immersed in them; and, since the divisions of the scale correspond to equal increments of volume immersed, it follows that the densities of the several liquids in which the instrument sinks to the successive divisions form a harmonic series.

Now, by increasing v, leaving W and V unchanged, we may increase the range of the instrument indefinitely.

Now let the solid be placed in the lower pan, care being taken that no bubbles of air remain attached to it, and let w i be the weight now required in the scale pan..

This weight will exceed w in consequence of the water displaced by the solid, and the weight of the water thus displaced will be wi-w, which is therefore the weight of a volume of water equal to that of the solid.

The instrument when unloaded serves for the range from 74 to 47 over proof; when loaded with the first weight it indicates from 46 to 13 over proof, with the second weight from 13 over proof to 29 under proof, and with the third from 29 under proof to pure water, the graduation corresponding to which is marked W at the bottom of the fourth scale.

His work (` w i lacxa) in twenty-four books, written in Greek, is rather a number of monographs than a connected history.

The only consonants are k,1, m, n and p, which with the gently aspirated h, the five vowels, and the vocalic w, make up all the letters in use.

Eventually this alphabet was enlarged (probably before the end of the 7th century) by the inclusion of two Runic letters for th and w.

The badge of the order is, with variations for the different classes, a white enamelled Danish cross with red and gold borders, bearing in the centre the letter W (V) and on the fourarms the inscription Gud og Kongen (For God and King).

The badge is a blue enamelled cross dependent from a lion surmounted by the ducal crown; the angles of the cross are filled by crowned W's and the centre bears the arms of Brunswick, a crowned pillar and a white horse, between two sickles.

For instance, 237578 w was printed @ 5070 8 3D; and the fact that Stevinus meant those encircled numerals to denote mere exponents is evident from his employing the very same sign for powers of algebraic quantities, e.g.

These are usually divided into three groups, the Central (the watershed between the two passes named), the Western or French, and the Eastern or Italian; in the following lists the initials " C," " W," and " E " show to which group each peak and pass belongs.

Quecees de Tignes (Val d'Isere to Termignon), snow (W).

The Eocene has altogether w Alps.

The tilting working chamber is connected with the stationary ports L and L' by means of the loose water-cooled joint W in Campbell's system, which is here shown.

It appears that the apostle had two names, each existing in a double form - Greek and Hebrew, Symeon (iivr,w) which was Graecized according to the sound into Simon, and Cephas (s;') which was Graecized according to the meaning into Peter (Mrpos).

Among other works of importance he wrote W o lag das Paradies?

This page gives an overview of all articles in the 1911 Brittanica which are alphabetized under W to Wan.

Among other of his works may be mentioned the short stories, Junge Liebe (1878), Salle Geschichten (1880), and the novels Moschko von Parma (1880), Ein Kampf ums Recht (1882), Der Prtisident (1884), Judith Trachtenberg (1890), Der W ahrheitsucher (5894).

When the stand on which w rests is shaken, a multiplied representation of this movement takes place at h, and any small body resting on that point, as for example a small screw s standing on its head, may be caused to topple over.

If the loaded rod is elastic its lower end may be fixed in a stand, and the spherically curved base w is no longer required.

On this point much recent with evidence has been collected, and it appears certain that w Sea.

Wadstein, Die eschatologische Ideengruppe, Antichrist, Weltsabbat, Weltende and Welgericht (Leipzig, 1896), which contains excellent material for the history of the idea in the West during the middle ages; W Meyer, "Ludus de Antichristo," in Sitzbericht der Miinchener Akad.

Thus Greek words are transliterated, as " chedrio " from KEBp6 w, " heremus "from €pmios; Greek idioms are reproduced, as " usque nos duci captivos," _ fws rou npas aixuaXwrw05vac, and retranslation into Greek is frequently necessary in order to correct the misrenderings of the translator or the corruptions already inherent in the Greek.

W, wax-yielding surface, covering true gland; s, septem, or carina; wh, webbed hairs.

The wheel W is therefore mounted on a guided rod, which is forced upwards by suitable levers and weights, and this relief of pressure is precisely proportional to the pressure on the respective bearings.

See 67a 1 a I"avrwv r&w aroµcov.

Clifford makes use of a quasi-scalar w, commutative with quaternions, and such that if p, q, &c., are quaternions, when p-I-wq= p'+wq', then necessarily p= p', q = q'.

In octonions the analogue of Hamilton's vector is localized to the extent of being confined to an indefinitely long axis parallel to itself, and is called a rotor; if p is a rotor then wp is parallel and equal to p, and, like Hamilton's vector, wp is not localized; wp is therefore called a vector, though it differs from Hamilton's vector in that the product of any two such vectors wp and coo- is zero because w 2 =o.

Let n, w be two quasi-scalars such that r t e =n, con = w, nw =w 2 = o.

Combebiac does not use K; and in place of, n he uses, u =1 7 - E, so that, u 2 = I, wµ = - µw = w, w 2 = o.

This relation cannot be directly integrated, so as to obtain the equation for the saturation-pressure, unless L and v - w are known as functions of 0.

Omitting w and neglecting the small variation of the specific heat of the liquid, the result is simply the addition of the term (c-b)/V to formula (23) log p=A+B/B - I - C log B-f-(c-b)IV..

The most important silicotungstic acids are silicodecitungstic acid H8W10S1036.3H20, tungstosilicic acid, H 8 W 12 SiO 42.20H 2 O, and silicoduodecitungstic or silicotungstic acid, H8W12S1042.29H20.

Ammonia does not react with tungsten or the dioxide, but with trioxide at a red heat a substance of the formula W 5 H 6 N 3 0 5 is obtained, which is insoluble in acids and alkalis and on ignition decomposes, evolving nitrogen, hydrogen and ammonia.

It therefore made the aspirates A, E, Q and the semi-vowel I into vowels, and apparently converted the semi-vowel Y = w into the vowel which it placed at the end of the alphabet and substituted for it as the sixth symbol of the alphabet the letter F with the old value of w.

In the Phoenician alphabet Zain was the seventh letter, occupying the same position and having the same form approximately (i) as the early Greek Z, while in pronunciation it was a voiced s-sound; Samech () followed the 'symbol for n of and was the ordinary s-sound, though, as we have seen, e it is in different Greek states at the earliest period as well as E; after the symbol for p came Zade (v), which was a strong palatal s, though in name it corresponds to the Greek Nra; while lastly Shin (W) follows the symbol for r, and was an sh-sound.

This would bring it into connexion with the Phoenician W (Shin), which, turned through a right angle, is possibly the Greek, though some forms of Zade on old Hebrew coins and gems equally resemble the Greek letter.

But the Phoenician form corresponding to it is the consonant w, and occupies the of position of the Greek digamma as sixth in the series.

The development of symbols for the long vowels o and w was also the work of the Ionians.

Centuries passed, however, before this symbol was generally adopted, Athens using only 0 for o, w and ov, the spurious diphthong, until the adoption of the whole Ionic alphabet in 403 B.C.'

The value of F in the Greek alphabet is w and not f as in Latin.

The strongest argument for the derivation from the Latin alphabet is undoubtedly the value of f attaching to P; for, as we have seen, the Greek value of this symbol is w, and its value as f arises only by abbreviation from FH.

The relations between this force P, the gravity W of the body, and the reaction S of the plane are then determined by a triangle of forces HKL.

If PQ be a short segment containing an isolated load W, we have FeFi.=W, MQ=MP; (3) hence F is discontinuous at a p concentrated load, diminishing by an amount equal to the load as we pass the loaded point to the right, whilst M is continuous.

The load on an element bx of the beam may be represented by wbx, where - w is in general a function of x.

As regards the funicular diagram, let LM be the line on which the pairs of corresponding sides of the two polygons meet, and through it draw any two planes w, w.

Each extraneous force W acting on a bar may be replaced (in an infinite number of ways) by two components P, Q in lines through the centres of the pins at the extremities.

In practice the forces W are usually vertical, and the components P, Q are then conveniently taken to be vertical also.

Again, any plane w is the locus of a system of null-lines meeting in a point, called the null-point of c. If a plane revolve about a fixed straight line p in it, its ntill-point describes another straight line p, which is called the conjugate line of p. We have seen that the wrench may be replaced by two forces, one of which may act in any arbitrary line p. It is now evident that the second force must act in the conjugate line p, since every line meeting p, p is a null-line.

As a simple example, take the case of a light frame, whose bars form the slides of a rhombus ABCD with the diagonal BD, suspended from A and carrying a weight W at C; and let it be required to find the stress in BD.

For this purpose the infinitesimal displacements of W the various joints are replaced by finite lengths proportional to them, and there- FIG.

If the string be uniform, w is constant, and T=wy +const.= w(yyi), (4)

If T be the tension, W the total weight, k the sag in the middle, and FIG.

Hence w, the weight per foot, varies as T, and we may write T =wX, where A is a constant length.

Now consider the tangent plane w at any point P of a confocal, the tangent plane fii at an adjacent point N, and a plane of through P parallel to of.

Obviously OV is parallel to the tangent to the path atP, and its magnitude is ds/dt, where s is the arc. If we project OV on the co-ordinate axes (rectangular or oblique) in the usual manner, the projections u, v, w are called the component velocities parallel to the axes.

If N=o, w is constant.

The motion of a rigid body in the most general case may be specified by means of the component velocities u, v, w of any point 0 of it which is taken as base, and the component angular velocities p, q, r.

Now T = 3/41w1, where w is the angular velocity and I is the moment of inertia about the instantaneous axis.

Since w varies as p, it follows that OH is constant, and the tangent plane at J is therefore fixed in space.

When, in any problem, the values of u, v, w, p, q, r have been determined as functions of t, it still remains to connect the moving axes with some fixed frame of reference.

The pulley or sheave FG has a weight W

Required the relation between the velocity of translation 02 of W and the angular velocity af of the differential barrel.

If the cord be fixed to the framework at the point B, instead of being wound on a barrel, the velocity of W is half that of AF.

In drawing these polygons the magnitude of the vector of the type Wr is the product Wr, and the direction of the vector is from the shaft outwards towards the weight W, parallel to the radius r.

A more accurate though still approximate expression for the force on the frame due to the acceleration of the piston whose weight is W is given by w2r cos 0 + r cos 20

To take a simple case, suppose a shaft supported on two bearings to carry a disk of weight W at its centre, I and let the centre of gravity of the disk be at a distance e from the axis of rotation, this small distance being due to imperfections of material or faulty construction.

Let OH=z; let W be the weight of the bob, F its centrifugal force.

By equation (72) we have E = w(v,2 vi2)/ag which, being divided by V=1/2(v,+vi), gives E/V=w(vfvi)/g; and consequently V2 in = gE/Vw (73)

Actual Energy of a Shifting Body.The energy which must be exerted on a body of the weight w, to accelerate it from a state of rest up to a given velocity of translation v, and the equal amount of work which that body is capable of performing by overcoming resistance while being retarded from the same velocity of translation v to a state of rest, is wvfI2g.

Let a small body of the weight w undergo translation in a circulai path of the radius p, with the angular velocity of deflexion a, so that the common linear velocity of all its particles is v=ap. Then the actual energy of that body is WV2/2g = Waip2/2g.

W and I being respectively the weight and the moment of inertia of the link about the axis through the Centre of gravity.

The later "c-w-µ€v was at first a solecism, an attempt to conjugate a " verb in µ.c " like the " verbs in w."

She was president of the National W oman Suffrage Association in 1865-1890.

In classical Latin its use is confined to the cases where, as in English quill, &c., the u is pronounced as w before a following vowel, but in old Latin it is found also in other combinations.

Many languages find the combination qu, when both sounds are consonantal (qw), difficult; q being the deepest guttural while u (English w) is a lip sound, the points of production are nearly as far separate as they can be.

In 1643, when the full legal establishment of Presbytery had just been consummated, the assembly, asked by the English W parliament to arrange a league to be signed in both countries for the furtherance of reformed religion, agreed, but asked that the league should be a religious one.

H „ Light and represents a rise of about £I P „ Picked £3 G „ Grey £3 S „ Superior £4 W „ White £4 Z „ Zins Each additional mark means a rise in the price, but it must be understood that it is quite possible for a quality denoted by two letters to be more valuable than one indicated by three or more, since every mark has not the same value.

To Photius we are indebted for almost all w possess of Ctesias, Memnon, Conon, the lost books of Diodorus Sictlus, and the lost writings of Arrian.

Similarly initial v became gw, as in gwin, from Latin vinum, remaining between vowels, though now written w, as in ciwed from civitas.

Adjectives having y or w are made feminine by a-affection, due to the lost feminine ending -a; thus gwyn, " white," fem.

Only second in importance to the above are W.'

Let two spherical pith balls of radius r and weight W, covered with gold-leaf so as to be conducting, be suspended by parallel silk threads of length 1 so as just to touch each other.

Observation of 0 with measurement of the value of 1 and r reckoned in centimetres and W in grammes gives us the potential difference of the balls in absolute C.G.S.

The movable plate can be thus depressed by placing on it a certain standard weight W grammes.

If W is the weight required to depress the attracted disk into the same sighted position when the plates are unelectrified and g is the acceleration of gravity, then the difference of potentials of the conductors tested is expressed by the formula V - V'=(d - d') /87 W where S denotes the area of the attracted disk.

It may well be doubted, however, whether his own extravagant desire for military glory was not equally injurious to his W country.

The interval O'O", termed the astigmatic difference, increases, in general, with the angle W made by the principal ray OP with the axis of the system, i.e.

For N to be constant for all values of w, a' tan w'/ a tan w must also be constant.

Fath Ali Shah undertook, at the outset of ith his reign, a contest with Russia on the western side, of War,w the Caspian, which became constant and harassing Russ a.

Israel or distinguished Israelites, the root being the same as in Jeshurun; (2) that Jashar (" lc) is a transposition of shir C, song); (3) that it should be pointed Yashir (W, sing; cf.

Heraclitus offers no analogy to the doctrine of four (not three) elements as different grades of tension; to the conception of fire and air as the " form," in Aristotelian terminology, of particulars; nor to the function of organizing fire which works by methodic plan to produce and preserve the world (irup i&w 1 3aSii'ov iri ')4vEru Nor, again, is there any analogy to the peculiar Stoic doctrine of universal intermingling (Kpavms Si iiXov).

Let the plane of the paper be a normal section of the surface of the stratum at the point B, making an angle w with the FIG.

If the current-function of the water referred to the body considered as origin is Ili, then the equation of the form of the crest of a wave of velocity w, the crest of which travels along with the body, is d =w ds where ds is an element of the length of the crest.

To integrate this equation for a solid of given form is probably difficult, but it is easy to see that at some distance on either side of the body, where the liquid is sensibly at rest, the crest of the wave will approximate to an asymptote inclined to the path of the body at an angle whose sine is w/V, where w is the velocity of the wave and V is that of the body.

If the angle between these is 20, the velocity of the body is w sec 0, where w for water is about 23 centimetres per second.

Modern critics are of opinion that, if genuine, it is an abridgment of a larger work by him (IIepi B &w).

The Czech (Cech) alphabet is the same as the English, with the omission of the letters q, w and x.

Kdnigs, Ber., 1879, 12, p. 2341) by passing a mixture of acetylene and hydrocyanic acid through a red-hot tube (W.

It was not until two centuries later that the historikos, the reciter of stories, superseded the historeon (to-Top&w), the seeker after knowledge.

In order to make the problem of flight more intelligible, the lever formed by the wing is prolonged beyond the body (b), and to the root of the wing so extended the weight (w, w') is attached; x represents the universal joint by which the wing is attached to the body.

When the wing ascends as shown at p, the air (fulcrum f) resists its upward passage, and forces the body (b) or its representative (w) slightly downwards.

When the wing descends as shown at p', the air (fulcrum f) resists its downward passage, and forces the body (b) or its representative (w') slightly upwards.

From this it follows that when the wing rises the body falls, and vice versa - the wing describing the arc of a large circle (f f), the body (b), or the weights (w, w') representing it, describing the arc of a small circle.

Under the Borough Funds Act 1872 the urban district council may, if in their judgment it is expedient, promote or oppose any local and personal bill or bills in parliament, or may Bills In prosecute or defend any legal proceedings necessary for the promotion or protection of the interests of the district, Parlla- and may charge the costs incurred in so doing to the w ent and rates under their control.

This distinction between Logos as ratio and Logos as oratio, so much used subsequently by Philo and the Christian fathers, had been so far anticipated by Aristotle's distinction between the g w Xo'yos and the Xoyos iv rff Ox?j.

The insurrection in Russo- Bulgaria, with its accompanying horrors, followed by Turkish the deposition of sultan Murad and the succession of w ar of 1877-78.

W Ilmmmmllullmmuuu11 ?. ?11?h.111llhnu1111111m1111111h011111?.11?111t1 Itllllln 1 t11111?11a.?111IIhllln111 1 1'1 1111 111 11 III 111 111 11 1111111111 11 11111111111111111111111111111u1111111111 e uulmm?m11unummmlln?

If 0 be the angle of tilt of the machine, and the other symbols be as before, it may be shown that the additional weight, w FIG.

When 0 is negative, as it is when the goods end of the machine is lower than the weights end, the value of w may be very appreciable.

With " vibrating " machines the value of in is in general so extremely small that w is of no practical importance in either of the above cases.

Using the symbols of the diagram, it can be shown that the effect of placing the weight W at E instead of F is to cause the end of the beam to descend, as if under the action of an additional weight, w, at F such that w Wa(ml - l +tan 0)/h.

The condition that must exist in order that the balance may weigh correctly for all positions of the weight W is w = o, or tan 0 = that is, the stay KG must be adjusted parallel to the line joining the points A and C. From the equation for w, it is seen that the larger h is the smaller w will be.

Thus in the case of a spring for which the values of the quantities were W =7 lb, n = 51, R = .30 in., .038 in., E = io,50o,000, the formula gives extension =1.764 in., while direct experiment gave extension =1.75 in.

And with a very long and weak spring for which the values of the quantities were W =1 oz., /1=233, R = 35 in., 0085 in., E = io,50o,000, the formula gives extension =22.78 in., while direct experiment with the spring gave 23.5 in.

The rim is also subject to a centrifugal tension of amount wv 2 /g pounds per square inch of section, where w is the weight in pounds of a length of one foot of the pulley rim one square inch in section, and v is the velocity of the rim in feet per second.

The magnitude of the unbalanced force, for a mass of w pounds at a radius of r feet and a velocity of v feet per second, is expressed by wv 2 /gr lb; and, since the force varies as the square of the velocity, it is necessary carefully to balance a pulley running at a high speed to prevent injurious vibrations.

For a single pulley of diameter D, turning on a fixed pin of diameter d, the relation of the effort E to the load W, where f is the coefficient of friction, is expressed by E/W = (D-pfd)/(D - fd) _ 1 +2fd/D approximately.

In particular, Plato taxes Parmenides with his inconsistency in attributing (as he certainly did) to the fundamental unity extension and sphericity, so that "the worshipped dv is after all a pitiful j.) " (W.

Thus 3Y0 3 0 - can be written 153 and 1 1 -6 o i W o can be written.

W&t Tyler The mob which had gathered at Maidstone and Canterbury marched on the capital many thousands strong, headed by a local demagogue named Wat Tyler, whom they had chosen as their captain; his most prominent lieutenant was the preacher John Ball.

A convenient form of measuring vessel is that devised by W Hempel.

The hot water from B, freed from tar, is pumped into a third tower C, through which cold air is forced by means of a Root's blower by the pipe w.

Below the entry W of the air-blast there is a throttle valve d which is closed as soon as the damper S opens the gas canal; thus a second security against the production of a mixture of air and gas is afforded.

The copious additional information given by later writers is all by way either of interpretation of local legends in the light of Ephorus's theory, or of explanation of the name "Pelasgoi"; as when Philochorus expands a popular etymology "stork-folk" (w€Xaa'yoi-- it €Xap'yoi) into a theory of their seasonal migrations; or Apollodorus says that Homer calls Zeus Pelasgian "because he is not far from every one of us," 6TL Tiffs ryes 7rEXas EaTCV.

Dorner's earlier view); impersonal pre-existence of the Logos, who became personal - compare and contrast Marcellus of Ancyra - at the Incarnation (W.

Although Royalists formed but a petty fraction of the majority, they raised the alarm that Mth i t w as seeking to restore monarchy and undo the work of the /8th g y of the Revolution.

The commercial grades are numerous, ranging by letters from A, the darkest, to N, extra pale, - superior to which are W, "window glass," and WW, "water white" varieties, the latter having about three times the value of the common qualities.

Gin, L'Electricien, 1903, 2 5, p. 5); and by the electrolysis of vanadium trioxide when heated in an evacuated glass tube (W.

Between 1 774 and 1789 he built scores of of ft ., the optical excellence of which approved itself W illiam by a crowd of discoveries.

A beam of sunlight admitted into a darkened room through a narrow aperture, and there dispersed into a vario-tinted band by the interposition of a prism, is not absolutely W continuous.

Since H' P = F 0, = y, from the focal length of the simple microscope, the visual angle w' is given by tan w'/y=I/f'=V, (I) in which f', = H' F', is the image-side focal length (see Lens).

Consequently the number of diameters will be N = tan w'/tan w = y/f.

The apparent size of the object seen through the lens is then tan w' = y/f.

The image viewed through the eyepiece appears then to the observer under the angle w", and as with the single microscope tan w" = I /f 2 ' (4) where f' 2 is the image-side focal length of the eyepiece.

If we replace y' in equation (4) by the value given by (3), we obtain tan w"/ y i/f2"=V, (5) the magnification of the complete microscope.

If this value of y be inserted in equation (5), we obtain the magnification number of the compound microscope N =tan w"/ tan w =Ol/f i 'f 2 ' =Vl.

In Attic Greek 0 and 12 were not really a pair, for o + o became not w but ov, o being a close and w an open sound.

This involves the expenditure of a quantity of work W, the amount in any particular case being found by the equation W = Q2 - Q I, where W is the work, expressed by its equivalent in British thermal units; Q2 the quantity of heat, also in B.Ther.U., given out at the higher temperature T2; and Q i the heat taken in at the lower temperature T1.

It is evident that the discharged heat Qz is equal to the abstracted heat Q ', plus the work expended, seeing that the work W, which causes the rise in temperature from T 1 to Tz, is the thermal equivalent of the energy actually expended in raising the temperature to the level at which it is rejected.

In conjunction with his elder brother he published in 1825 a well - known treatise on waves, Die W ellenlehre auf Experimente gegrundet; and in 1833 he collaborated with his younger brother, the physiologist Eduard Friedrich Weber (1806-1871), in an investigation into the mechanism of walking.

After some work, they found there were nineteen different letters used, with "E, I, L, O, U, Q, and W" missing.