X Sentence Examples

The words ascribed to Christ in Luke x.

It will be seen that (1) the increase in equivalent volume is about 6.6; (2) all the topic parameters are increased; (3) the greatest increase is effected in the parameters x and tG, which are equally lengthened.

The swellings have been found to be due to a curious hypertrophy of the tissue of the part, the cells being filled with an immense number of minute bacterium-like organisms of V, X or Y shape.

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

Two relations R and R' are said to be ordinally similar, if a one-one relation holds between the members of the two fields of R and R', such that if x and y are any two members of the field of R, such that x has the relation R to y, and if x' and y are the correlates in the field of R' of x and y, then in all such cases x has the relation R' to y', and conversely, interchanging the dashes on the letters, i.e.

Multiplying out the right-hand side and comparing coefficients X1 = (1)x1, X 2 = (2) x2+(12)x1, X3 = (3)x3+(21)x2x1+ (13)x1, X4 = (4) x 4+(31) x 3 x 1+(22) x 2+(212) x2x 1 +(14)x1, Pt P2 P3 P1 P2 P3 Xm=?i(m l m 2 m 3 ...)xmlxm2xm3..., the summation being for all partitions of m.

In the very early rock inscriptions of Thera (700-600 B.C.), written from right to left, it appears in a form resembling the ordinary Greek X; this form apparently arose from writing the Semitic symbol upside down.

To reduce these figures to a common standard, so that the volumes shall contain equal numbers of molecules, the notion of molecular volumes is introduced, the arbitrary values of the crystallographic axes (a, b, c) being replaced by the topic parameters' (x, ?i, w), which are such that, combined with the axial angles, they enclose volumes which contain equal numbers of molecules.

Assuming then 01 to have the coefficients B1, B2,...B,, and f l the coefficients A 1, A21...A,n, we may equate coefficients of like powers of x in the identity, and obtain m+n homogeneous linear equations satisfied by the m+n quantities B1, 2, ...B n, A 1, A 2, ...A m.

In the case of the briquette the position of the foot of the ordinate u is expressed by co-ordinates x, y, referred to a pair of axes parallel to a pair of sides of the base of the briquette.

Coote's Remarkable Maps of the X Vth, X Vlth and X VIIth Centuries reproduced in their Original Size (Amsterdam, 1894-1897), and Bibliotheca lindesiana (London, 1898) with facsimiles of the Harleian and other Dieppese maps of the 16th century.

According to this notation, the three equations of motion are dt2 = b2v2E + (a2 - b2) d.s dt =b2v2rj+(a2 - b2) dy d2 CIF - b2p2+(a2_b2)dz It is to be observed that denotes the dilatation of volume of the element situated at (x, y, z).

In the a x = aixi+a2x2, observe that a a = a2, ab = aibi +a2b2.

No terms in x or x 2 need be considered.

The distance f i, which the actual focal length must exceed, is given by d (f1 2 R2) x; so that f1 = 2 R2/X (1) Thus, if X = p j, R= i ?, we find f1= 800 inches.

On either side of any one of them the illumination is distributed according to the same law as for the central image (m = o), vanishing, for example, when the retardation amounts to (mn t 1)X.

If BQ be the direction for the first minimum (the darkness between the central and first lateral band), the relative retardation of the extreme rays is (mn+1)X.

The aperture of the unretarded beam may thus be taken to be limited by x = - h, x = o, y= - 1, y= +1; and that of the beam retarded by R to be given by x =o, x =h, y = - 1, y = +l.

For the retarded stream the only difference is that we must subtract R from at, and that the limits of x are o and +h.

When a, b, X are regarded as constant, the first factor may be omitted, - as indeed should be done for consistency's sake, inasmuch as other factors of the same nature have been omitted already.

Let x, y, z be the co-ordinates of any particle of the medium in its natural state, and, 7 7, the displacements of the same particle at the end of time t, measured in the directions of the three axes respectively.

Since the dimensions of T are supposed to be very small in com d parison with X, the factor dy (--) is sensibly constant; so that, if Z stand for the mean value of Z over the volume T, we may write TZ y d e T ?

We will now apply (18) to the investigation of a law of secondary disturbance, when a primary wave = sin (nt - kx) (19) is supposed to be broken up in passing the plane x = o.

Note that these phagocytic cells are pushing out protoplasmic processes (pseudopodia) by which they grasp their victims. (X woo diam.) FIG.

These vesicles are filled with the colloid material (x 90 diam.) J FIG.

Brussels was again the headquarters in 1741, by which time Voltaire had finished the best and the second X XVIII.

It is not probable that the sweet-smelling gums and resins of the countries of the Indian Ocean began to be introduced into Greece before the 8th or 7th century B.C., and doubtless XiOavos or X q /3avw-rOs first became an article of extensive commerce only after the Mediterranean trade with the East had been opened up by the Egyptian king Psammetichus (c. 664-610 B.C.).

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.

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.

These were "lost sheep of the house of Israel"; but Christ's freedom from Jewish exclusiveness is also brought out (I) as regards Samaritans, by the rebuke administered to the disciples at ix.52 sqq., the parable in x.

In the second of these passages the disciples are exhorted to choose a life of voluntary poverty; the nearest parallel is the ideal set before the rich young man at Mark x.

The cartesian equation referred to the axis and directrix is y=c cosh (x/c) or y = Zc(e x / c +e x / c); other forms are s = c sinh (x/c) and y 2 =c 2 -1-s 2, being the arc measured from the vertex; the intrinsic equation is s = c tan The radius of curvature and normal are each equal to c sec t '.

Thus water being about Boo times denser than air and mercury 13.6 times denser than water, k/h = 6,/p = 800 X 13.6 = Io,880; (2) and with an average barometer height of 30 in.

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.

Thus if the plane is normal to Or, the resultant thrust R =f fpdxdy, (r) and the co-ordinates x, y of the C.P. are given by xR = f f xpdxdy, yR = f f ypdxdy.

As a rule these equations are established immediately by determining the component acceleration of the fluid particle which is passing through (x, y, z) at the instant t of time considered, and saying that the reversed acceleration or kinetic reaction, combined with the impressed force per unit of mass and pressure-gradient, will according to d'Alembert's principle form a system in equilibrium.

Taking the axis of x for an instant in the normal through a point on the surface H = constant, this makes u = o, = o; and in steady motion the equations reduce to dH/dv=2q-2wn = 2gco sin e, (4) where B is the angle between the stream line and vortex line; and this holds for their projection on any plane to which dv is drawn perpendicular.

In the equations of uniplanar motion = dx - du = dx + dy = -v 2 ?, suppose, so that in steady motion dx I +v24 ' x = ?'

When the cylinder r =a is moved with velocity U and r =b with velocity U 1 along Ox, = U b e - a,1 r +0 cos 0 - U ib2 - 2 a, (r +Q 2 ') cos 0, = - U be a2 a2 (b 2 - r) sin 0 - Uib2 b1)a, (r - ¢2 sin 0; b and similarly, with velocity components V and V 1 along Oy a 2 b2 ?= Vb,_a,(r+r) sin g -Vi b, b2 a, (r+ 2) sin 0, (17) = V b, a2 a, (b2 r) cos 0+Vi b, b, a, (r- ¢ 2) cos h; (18) and then for the resultant motion z 2zz w= (U 2 + V2)b2a a2U+Vi +b a b a2 U z Vi -(U12+V12) b2 z a2b2 Ui +VIi b 2 - a 2 U1 +Vii b 2 - a 2 z The resultant impulse of the liquid on the cylinder is given by the component, over r=a (§ 36), X =f p4 cos 0.ad0 =7rpa 2 (U b z 2 + a 2 Uib.2bz a2); (20) and over r =b Xi= fp?

Then, if the outside cylinder is free to move - 2a2 2 X 1 = 0, T T = b2 a2, 7rpa 2 Ub 2+ a 2.

Taking two planes x = =b, and considering the increase of momentum in the liquid between them, due to the entry and exit of liquid momentum, the increase across dy in the direction Oy, due to elements at P and P' at opposite ends of the diameter PP', is pdy (U - Ua 2 r2 cos 20 +mr i sin 0) (Ua 2 r 2 sin 2 0+mr 1 cos 0) + pdy (- U+Ua 2 r 2 cos 2 0 +mr1 sin 0) (Ua 2 r 2 sin 2 0 -mr 1 cos 0) =2pdymUr '(cos 0 -a 2 r 2 cos 30), (8) and with b tan r =b sec this is 2pmUdo(i -a 2 b2 cos 30 cos 0), (9) and integrating between the limits 0 = 27r, the resultant, as before, is 27rpmU.

For the liquid filling the interior of a rotating elliptic cylinder of cross section x2/a2+y2/b2 = 1, /(4) = m l (x 2 / a2 - - y2 /b 2) (5) with V21G1' =-2R =-2 m 1 (I / a2 + 21b2), 214 = m l (x2 / a2 + y2 / b2) - IR(x2+y2) = I R (x2 - y2) (a 2 - b2)/(a2+b2), cp 1 = Rxy (a 2 - b2)/(a2 +b2), w1 = cb1 +% Pli = - IiR(x +yi)2(a2b2)/(a2+b2).

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.

A relative stream line, along which 1/,' = Uc, is the quartic curve y-c=?![2a(r-x)], x = 4a2y2-(y g)4, r- 4a2y2 +(y c) 4, 7) 4 a (y-c) 4a(y and in the absolute space curve given by 1', dy= (y- c)2, x= 2ac_ 2a log (y -c) (8) 2ay y - c 34.

Similarly, with the function (19) (2n+ I) 3 ch (2n+ I) ITrb/a' (2) Changing to polar coordinates, x =r cos 0, y = r sin 0, the equation (2) becomes, with cos 0 =µ, r'd + (I -µ 2)-d µ = 2 ?-r3 sin 0, (8) of which a solution, when = o, is = (Ar'+) _(Ari_1+) y2,, ?

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.

If there are more B corners than one, either on xA or x'A', the expression for i is the product of corresponding factors, such as in (5) Restricting the attention to a single corner B, a = n(cos no +i sin 110) _ (b-a'.0-a) +1!

Ja - u  ?I a -a b -u' sh nS2=sh log (Q)=?a - b a - a' b - u' At x where = co, u = o, and q= go, (O n b - a ' a + a -b a' cio) - ?a-a'?b a-a' q In crossing to the line of flow x'A'P'J', b changes from o to m, so that with q = Q across JJ', while across xx the velocity is qo, so that i n = go.

Generally, by making a' = -oo, the line x'A' may be taken as a straight stream line of infinite length, forming an axis of symmetry; and then by duplica tion the result can be ob A tained, with assigned n, a, and b, of the efflux from a symmetrical converging FIG.

For a cavity filled with liquid in the interior of the body, since the liquid inside moves bodily for a motion of translation only, 41 = - x, 42 = -, 43 = - z; (2) but a rotation will stir up the liquid in the cavity, so that the'x's depend on the shape of the surface.

Over the ellipsoid, p denoting the length of the perpendicular from the centre on a tangent plane, px _ pv _ _ pz 1= a2+X' b +A' n c2+A p2x2 + p2y2 p2z2 I (a2 - + X)2 (b 2 +x)2 + (0+X)2, p 2 = (a2+A)12+(b2+X)m2+(c2+X)n2, = a 2 1 2 +b 2 m 2 +c 2 n 2 +X, 2p d = ds; (8) Thence d?

The continuity is secured if the liquid between two ellipsoids X and X 11 moving with the velocity U and 15 1 of equation (II), is squeezed out or sucked in across the plane x=o at a rate equal to the integral flow of the velocity I across the annular area a l.

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'.

Between two concentric spheres, with a 2 +A = r 2, a2+A1=a12, A=B =C =a3/3r3, a 3 a 3 a3 _ a3 Cb _1U x r 3 a13 Y'=2 Uy2 r3 3 a13 .

The extension to the case where the liquid is bounded externally by a fixed ellipsoid X= X is made in a similar manner, by putting 4 = x y (x+ 11), (io) and the ratio of the effective angular inertia in (9) is changed to 2 (B0-A0) (B 1A1) +.a12 - a 2 +b 2 a b1c1 a -b -b12 abc (Bo-Ao)+(B1-A1) a 2 + b 2 a1 2 + b1 2 alblcl Make c= CO for confocal elliptic cylinders; and then _, 2 A? ?

The determination of the O's and x's is a kinematical problem, solved as yet only for a few cases, such as those discussed above.

Conversely, if the kinetic energy T is expressed as a quadratic function of x, x x3, y1, y2, y3, the components of momentum, the partial differential coefficient with respect to a momentum component will give the component of velocity to correspond.

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 =..

When no external force acts, the case which we shall consider, there are three integrals of the equations of motion (i.) T =constant, x 2 +x 2 +x 2 =F 2, a constant, (iii.) x1y1 +x2y2+x3y3 =n = GF, a constant; and the dynamical equations in (3) express the fact that x, x, xs.

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.

Table of Rifling for Stability of an Elongated Projectile, x Calibres long, giving S the Angle of Rifling, and n the Pitch of Rifling in Calibres.

The components of force, X, Y, and N, acting on the liquid at 0, and reacting on the body, are then X=It.

The Book of Noah is mentioned in Jubilees x.

As for the author, he was no Essene, for he recognizes animal sacrifices and cherishes the Messianic hope; he was not a Sadducee, for he looks forward to the establishment of the Messianic kingdom (x.); nor a Zealot, for the quietistic ideal is upheld (ix.), and the kingdom is established by God Himself (x.).

B, Archigetes sieboldi; X 60 (from Leuckart).

B, head showing the suckers, proboscides and excretory canals; X 25.

The uterus (X in figure C) begins in all cases at the shell gland (c, d) and may exhibit a swelling (R S) for the retention of the spermatozoa..

C, invaginated head of Cysticercus cellulosae, showing the bent neck and receptacle r; X 30.

Zinc carbonate, ZnCO 3, occurs in nature as the mineral calamine (q.v.), but has never been prepared artificially, basic carbonates, ZnCO 3 .xZn(OH) 2, where x is variable, being obtained by precipitating a solution of the sulphate or chloride with sodium carbonate.

A queen of this people (the " Queen of Sheba ") is said (1 Kings x.) to have visited Solomon about 950 B.C. There is, however, no mention of such a queen in the inscriptions.

Addai was supposed to be one of the Seventy of Luke x.

This functionary is first formally mentioned under Leo X.(1513-1521) in the proceedings in connexion with the canonization of St Lorenzo Giustiniani.

A, Fasciola hepatica, from the ventral surface (X 2); the alimentary and nervous systems only shown on the left side of the figure, the excretory only on the right; a, right main branch of the intestine; c, a diverticulum; g, lateral ganglion; n, lateral nerve; o, mouth; p, pharynx; s, ventral sucker; cs, cirrus sac; d, left anterior dorsal excretory vessel; m, main vessel; v, left anterior ventral trunk; x, excretory pore.

The central nervous system (x) is highly developed, and in Loxosoma bears a pair o` eyes.

In its modern form the theorem, which is true for all values of n, is written as (x +a) n -1+ I.

Now in order to [do] this, it appeared that in all the series the first term was x; that the second terms R- x 3 3x 3, 3x 3, &c., were in arithmetical progression; and consequently that the first two terms of all the series to be interpolated would be x-- 3, x- `3, x-- 3, &c.

I enquired therefore how, in these series, the rest of the terms may be derived from the first two being given; and I found that by putting m for the second figure or term, the rest should be produced by the continued multiplication of the terms of this seriesI X m 2 I X m - 2 This rule I therefore applied to the series to be interpolated.

The hyperbolic or Gudermannian amplitude of the quantity x is ta n (sinh x).

Hence the density v is given by 47rabc (x2/a4+y2/b4-I-z2/c4), and the potential at the centre of the ellipsoid, and therefore its potential as a whole is given by the expression, adS Q dS V f r 47rabc r' (x2/a4-I-y2/b4+z2/c4) Accordingly the capacity C of the ellipsoid is given by the equation 1 I J dS C 47rabc Y (x 2 +y 2 + z2) V (x2/a4+y2/b4+z2/c4) (5) It has been shown by Professor Chrystal that the above integral may also be presented in the form,' foo C 2 J o J { (a2 + X) (b +X) (c 2 + X) } (6).

It has been shown above that the potential due to a charge of q units placed on a very small sphere, commonly called a point-charge, at any distance x is q/x.

The mathematical importance of this function called the potential is that it is a scalar quantity, and the potential at any point due to any number of point charges ql, q2, q3, &c., distributed in any manner, is the sum of them separately, or qi/xl+q2/x2+q3/x3+&c. =F (q/x) =V (17), where xi, x2, x 3, &c., are the distances of the respective point charges from the point in question at which the total potential is required.

Then the electric force due to the point s' charge q at distance x is q/x, and the resolved part normal to the element of surface dS is q cos0/x 2.

Let us apply the above theorem to the case of a small parallelepipedon or rectangular prism having sides dx, dy, dz respectively, its centre having co-ordinates (x, y, z).

Its angular points have then co-ordinates (x t Zdx, y t Zdy, z * zdz).

Let V be the potential at the centre of the prism, then the normal forces on the two faces of area dy.dx are respectively RI dx2 d xl and (dx 2 d x), dV d2 and similar expressions for the normal forces to the other pairs of faces dx.dy, dz.dx.

See La Fondation de la regence d'Alger, histoire des Barberousse, chronique arabe du X VI siecle published by Sander Rang and Ferdinand Denis, Paris, 1837 - for a curious Moslem version of their story.

I, 4), the modern Citta di Castello, he set up a temple at his own expense and adorned it with statues of Nerva and Trajan (x.

The width of each of the portions aghc and acfe cut away from the lens was made slightly greater than the focal length of lens X tangent of sun's greatest diameter.

Here, in order to fulfil the purposes of the previous models, the distance of the centres of the lenses from each other should only slightly exceed the tangent of sun's diameter X focal length of lenses.

To this the original author added as an appendix x.

Thirdly, the vials source from the time of Pompey (circa 63),- x.

The author wrote x.

In it the prophet receives a new commission, x.

Buchanan found a mean of 20 experiments made by piezometers sunk in great depths on board the " Challenger " give a coefficient of compressibility K=491 X 107; but six of these experiments made at depths of from 2740 to 3125 fathoms gave K=480Xio 7.

Feuillet des Conches Louis X VI., Marie Antoinette et Madame Elisabeth, lettres et documents inedits (6 vols., Paris, 1864-1873), while most of the works on Marie Antoinette published before the appearance of Arneth's publications (1865, &c.) are based partly on these forgeries.

X a.

The radial distance x is at any instant proportional to the force acting through the spring.

It appears tolerably safe to conclude that, whatever errors 'may have affected the determination, the diameter or distance of the particles of water is between the two thousand and the ten thousand millionth of an inch " (= between 125 X I o 8 and 025 X 10 -8 cms.).

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.

T =273, we obtain R =1.35 X Io -16 and this enables us to determine the mean velocities produced by heat motion in molecules of any given mass.

Plato (Rep. x.) embodies the idea in one of his finest myths.

The formula for length of scale is, length = sighting radius X tangent of the angle of elevation.

The finest of these apartments, containing beautiful arabesque x XVII.

Take any base X'X, and draw lines at right angles to this base through all the angular points of the figure.

Then, if we take ordinates Kb, Lg, Mc, Nd, Pf, equal to B'B, GG', C'C, D'D, FF', the figure abgcdfe will be the equivalent trapezoid, and any ordinate drawn from the base to the a LM N P e X top of this trapezoid will be equal to the portion of this ordinate (produced) which falls within the original figure.

Volume =height X area of end =length of edge X area of cross-section; the " height " being the perpendicular distance between the two ends.

B where x = the distance between the two lines = N /No.

These formulae also hold for converting moments of a solid figure with regard to a plane into moments with regard to a parallel plane through the centroid; x being the distance between the two planes.

Then we may, ignoring the units G and H, speak of ON and NP as being equal to x and u respectively.

Let KA and LB be the positions of NP corresponding to the extreme values of x.

To illustrate the importance of the mensuration of graphs, suppose that we require the average value of u with regard to x.

It arises mainly in statistics, when the ordinate of the trapezette represents the relative frequency of occurrence of the magnitude represented by the abscissa x; the magnitude of the abscissa corresponding to the median ordinate is then the " median value of x."

If there are m of these strips, and if the breadth of each is h, so that H =mh, it is convenient to write x in the form xo+Oh, and to denote it by x 0, the corresponding value of u being ue.

If the planes of one set divide it into m slabs of thickness h, and those of the other into n slabs of thickness k, so that H =mh, K = nk, then the values of x and of y for any ordinate may be denoted by xo+Oh and yo+Ok, and the length of the ordinate by uo, 0.

The area of the trapezette, measured from the lower bounding ordinate up to the ordinate corresponding to any value of x, is some function of x.

In the notation of the integral calculus, this area is equal to f x o udx; but the notation is inconvenient, since it implies a division into infinitesimal elements, which is not essential to the idea of an area.

In the same way the volume of a briquette between the planes x = xo, y = yo, x= a, y = b may be denoted by [[Vx,y ]y=yo] u 'x' =xo.

The statement that the ordinate u of a trapezette is a function of the abscissa x, or that u=f(x), must be distinguished from u =f(x) as the equation to the top of the trapezette.

If this is done for every possible value of x, there will be a series of ordinates tracing out a trapezette with base along OX.

Hence the volume of each element of the solid figure is to be found by multiplying the area of the corresponding element of the trapezette by 1, and therefore the total volume is 1 X area of trapezette.

The simplest case is that in which u is constant or is a linear function of x, i.e.

The next case is that in which u is a quadratic function of x, i.e.

Generally, if the area of a trapezette for which u is an algebraical function of x of degree 2n is given correctly by an expression which is a linear function of values of u representing ordinates placed symmetrically about the mid-ordinate of the trapezette (with or without this mid-ordinate), the same expression will give the area of a trapezette for which u is an algebraical function of x of degree 2n + 1.

This will be seen by taking the mid-ordinate as the ordinate for which x = o, and noticing that the odd powers of x introduce positive and negative terms which balance one another when the whole area is taken into account.

When u is of degree 4 or 5 in x, we require at least five ordinates.

Writing m = 2p, and grouping the coefficients of the successive differences, we shall find area = 2ph up+ 2 652up + 3 p4365p2 84up 3p,6 - 21p4 28p2 15120 If u is of degree 2f or 2f + i in x, we require to go up to b 2f u p, so that m must be not less than 2f.

The general formulae of § 54 (p being replaced (i) by 2m) may in the same way be applied to obtain formulae giving the area of the trapezette in terms of the mid-ordinates of the strips, the series being taken up to b 2f ul m or /th 2J ug m at least, where u is of degree 2f or 2f + I in x.

In the case of the sphere, for instance, whose radius is R, the area of the section at distance x from the centre is lr(R 2 -x 2), which is a quadratic function of x; the values of So, Si, and S2 are respectively o, 7rR 2, and o, and the volume is therefore s.

In the case of a pyramid, of height h, the area of the section by a plane parallel to the base and at distance x from the vertex is clearly x 2 /h 2 X area of base.

Let the distance between the parallel planes through AB and CD be h, and let a plane at distance x from the plane through AB cut the edges AC, up -f- .

By drawing Ac and Ad parallel to BC and BD, so as to meet the plane through CD in c and d, and producing QP and RS to meet Ac and Ad in q and r, we see that the area of Pqrs is (x/h - x 2 /h 2) X area of cCDd; this also is a quadratic function of x.

In the case of the parabolic trapezette, for instance, xu is of degree 3 in x, and therefore the first moment is lh(xouo+4xlui-+x2u2).

In the case, therefore, of any solid whose cross-section at distance x from one end is a quadratic function of x, the position of the crosssection through the centroid is to be found by determining the position of the centre of gravity of particles of masses proportional to So, S2, and 4S 1, placed at the extremities and the middle of a line drawn from one end of the solid to the other.

If u is an algebraical function of x of degree not exceeding p, and if the area of a trapezette, for which the ordinate v is of degree not exceeding p+q, may be expressed by a formula Aovo-1--yivi+..

To extend these methods to a briquette, where the ordinate u is an algebraical function of x and y, the axes of x and of y being parallel to the sides of the base, we consider that the area of a section at distance x from the plane x = o is expressed in terms of the ordinates in which it intersects the series of planes, parallel to y=o, through the given ordinates of the briquette (§ 44); and that the area of the section is then represented by the ordinate of a trapezette.

This ordinate will be an algebraical function of x, and we can again apply a suitable formula.

Suppose, for instance, that u is of degree not exceeding 3 in x, and of degree not exceeding 3 in y, that it contains terms in x3y3, x 3 y 2, x2y3, &c.; and suppose that the edges parallel to which x and y are measured are of lengths 2h and 3k, the briquette being divided into six elements by the plane x=xo+h and the planes y = yo+k, y = yo+2k, and that the 12 ordinates forming the edges of these six elements are given.

The area of the section by a plane at distance x from the edge 0 is a function of x whose degree is the same as that of u.

The volume of the briquette for which u is a function of x and y is found by the operation of double integration, consisting of two successive operations, one being with regard to x, and the other with regard to y; and these operations may (in the cases with which we are concerned) be performed in either order.

The first, which is the best known but is of limited application, consists in replacing each successive portion of the figure by another figure whose ordinate is an algebraical function of x or of x and y, and expressing the area or volume of this latter figure (exactly or approximately) in terms of the given ordinates.

This means that, if the minor trapezette consists of k strips, v will be of degree k or k - I in x, according as the data are the bounding ordinates or the mid-ordinates.

P2 = x2 I A I + x 2 I A I +.

The position of the central ordinate is given by x = v 1 /po, and therefore is given approximately by pl/po.

These results may be extended to the calculation of an expression of the form fxo u4(x)dx, where 0(x) is a definite function of x, and the conditions with regard to u are the same as in § 82.

The formulae can be adapted to the case in which cp(x) is tabulated for xl,.

The expressions in square brackets are in each case to be taken as relating to the extreme values x =xo and x=xm, as in §§ 75 and 76.

The formulae of § 82 can be extended to the case of a briquette whose top has close contact with the base all along its boundary; the data being the volumes of the minor briquettes formed by the planes x =x0, x = x i,

If the data of the briquette are, as in § 86, the volumes of the minor briquettes, but the condition as to close contact is not satisfied, we have y "`x P u dx dy = K + L + R - X111010-0,0 f xo yo i'?

Either or both of the expressions K and L will have to be calculated by means of the formula of § 84; if this is applied to both expressions, we have a formula which may be written in a more general form f f 4 u4(x, y) dx dy = u dx dy.

It is also clearly impossible to express u as an algebraical function of x and y if some value of du/dx or duldy is to be infinite.

One method is to construct a table for interpolation of x in terms of u, and from this table to calculate values of x corresponding to values of u, proceeding by equal intervals; a quadrature-formula can then be applied.

Even where u is an explicit function of x, so that f x udx may be expressed in terms of x, it is often more convenient, for construction of a table of values of such an integral, to use finite-difference formulae.

Each of the above formulae involves an arbitrary constant; but this disappears when we start the additions from 'a known value of X udx.

The formulae may be used for extending the accuracy of tables, in cases where, if v represents the quantity tabulated, hdv/dx or h 2 d 2 v /dx 2 can be conveniently expressed in terms of v and x to a greater degree of accuracy than it could be found from the table.

As Johnson thought it unsafe to pursue the routed army his victory had no other effect than the erection here of the useless defences of Fort William Henry, but as it was the only success in a year of gloom parliament rewarded him with a grant of X 5000 and the title of a baronet.

Both narratives are doubtless based upon actual occurrences - the cures narrated in Mark ii., iii., viii., x.

Mary and Martha are admittedly identical with the sisters in Luke x.

Thus a push or a compression of the X air is transmitted onwards in the direction OX.

There is also the " external " applied pressure X, and the total momentum flowing out per second is X-I-P4-W-1-p(U - u)2.

If then we apply a pressure X given by (5) at every point, and move the medium with any uniform velocity U, the disturbance remains fixed in space.

Or if we now keep the undisturbed parts of the medium fixed, the disturbance travels on with velocity U if we apply the pressure X at every point of the disturbance.

If the velocity U is so chosen that E - poU 2 = o, then X = o, or the wave travels on through the action of the internal forces only, unchanged in form and with velocity U = (E/p).

If, however, we put on external forces of the required type X it is obvious that any wave can be propagated with any velocity, and our investigation shows that when U has the value in (6) then and only then X is zero everywhere, and the wave will be propagated with that velocity when once set going.

In the momentum equation (4) we may now omit X and it becomes 0.+P(U - u) 2 =poU2.

Thus the second A = 2 X 2 X 2, and we may regard it as an ascent through two fifths in succession and then a descent through an octave.

The third 4= 5 X Z X z or ascent through an interval 4, which has no special name, and a descent through two octaves, and so on.

If we start with F as key-note, besides a small difference at we have as the fourth from it 3 X 4 = y, making with B = I R 5 an interval and requiring a new note, B flat.

Since U=n X where U is the velocity of sound, X the wave-length, and n the frequency, it follows that the forward frequency is greater than the backward frequency.

The simplest form of wave, so far as our sensation goes - that is, the one giving rise to a pure tone - is, we have every reason to suppose, one in which the displacement is represented by a harmonic curve or a curve of sines, y=a sin m(x - e).

If we put this in the form y=a sin (x - e), we see that y=o, for x =e, e +aX, e+2A, e+;X, and so on, that y is + from x=e to x=e+iX, - from e+zX to e+?A, and so on, and that it alternates between the values+a and - a.

If we transfer 0 to A, e=o, and the curve may be represented by y=a sin A x.

If now the curve moves along unchanged in form in the direction ABC with uniform velocity U, the epoch e =OA at any time t will be Ut, so that the value of y may be represented as 2 y=a sin T (x - Ut).

Fourier's theorem asserts that such a curve may be built up by the superposition, or addition of ordinates, of a series of sine curves of wave-lengths X, IX, 3A, 4A...

Now we may resolve these trains by Fourier's theorem into harmonics of wave-lengths X, 2X, 3A, &c., where X=2AB and the conditions as to the values of y can be shown to require that the harmonics shall all have nodes, coinciding with the nodes of the fundamental curve.

Putting A /M =n 2 the equation becomes x+n 2 x=o, whence x =A sin nt, and the period is 27r/n.

The result was an amplitude of 1.27 X 107 cm.

We may represent the displacement due to one of the trains by y l =a sin 2 i (24) where x is measured as in equation (16) from an ascending node as A in fig.

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 U is the velocity of sound in a gas at pressure P with density p, and if waves of length X and frequency N are propagated through it, then the distanc?e l between the dust-heaps is 2 = N - zN Vyp' where y is the ratio of the two specific heats.

We might make this investigation more general by introducing a force X as in the investigation for air, but it hardly appears necessary.

The middle of the string is a loop. In the next mode A and C represent the ends and AC = X = and n 2 = U/A 2 = 2U/2/ = (2/21) A / (T/m).

If then y=a sin (x - Ut),, t=2,ro-ra cos (x - Ut).

But keeping r/X small we may as before form stationary waves, and it is evident that the series of fundamental and overtones will be just as with the air in pipes, and we shall have the same three types - fixed at one end, free at both ends, fixed at both ends - with fundamental frequencies respectively 41, p ' 21 V p, and I velocity in rod =velocity in air X distance between dust heaps.

If the two interfering waves, being still of same length X, be in opposite phases, or sõ that one is in advance of the other by 2X, and consequently one produces in the air the opposite state of motion to the other, then the resultant wave is one of the same length X, but the excursions of the particles are decreased, being the difference between those due to the component waves as in fig.

If the difference of phase be varied gradually from zero to - X, the resulting sound will 2 gradually decrease from a maximum to a minimum.

The amplitudes of these tones are proportional to the products of a and b multiplied by X or µ.

But inasmuch as the successive orders are proportional to A X 2 A 3, or µµ 2 µ 3, and X and µ are small, they are of rapidly decreasing importance, and it is not certain that any beyond those in equation (35) correspond to our actual sensations.

We may see how this occurs by supposing that the restoring force of the receiving mechanism is represented by Ax- f-µx 2, where x is the displacement and µx 2 is very small.

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.

In connexion with the journey from this region to Jerusalem three striking incidents are recorded, x.

In one is represented Moses receiving the Old Law, in the other Christ delivers to St Peter the New Law - a charter sealed with the X P monogram.

In analytical geometry, the equation to the sphere takes the forms x 2 +y 2 +z 2 =a 2, and r=a, the first applying to rectangular Cartesian co-ordinates, the second to polar, the origin being in both cases at the centre of the sphere.

Two spheres intersect in a plane, and the equation to a system of spheres which intersect in a common circle is x 2 + y 2 + z 2 +2Ax -fD = o, in which A varies from sphere to sphere, and D is constant for all the spheres, the plane yz being the plane of intersection, and the axis of x the line of centres.

Maitland describes it (Political Theories of the Middle Ages, p. x.), have an essentially "statelike character."

Let a b be Ak X ?

The greatest moment CG at C is x(l-x)/l.

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.

If X ---- r - i the scales are so chosen i that a inch represents 1 x ' n - "--, in.

If the load rests directly on the main girder, the greatest -Iand - shears at C will be wXAGC and -w X CHB.

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

H = wL2/8y, or, calling x the distance from the vertex to the point of support, H = wx2/2y.

The value of R, the tension at any point at a distance x from the vertex, is obtained from the equation R 2 = H2 +V2 = w2x4 /4Y 2 +w2x2, or, 2.

R=wx (I+x2/4Y2) Let i be the angle between the tangent at any point having the co-ordinates x and y measured from the vertex, then 3..

The integration can be performed when M is expressed in terms of x.

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.

Then the deflection at the centre is the value of y for x = a, and is _ 5 wa4 S - 14 EI' The radius of curvature of the beam at D is given by the relation R=EI/M.

They pay no premium, and generally receive a salary of P8 to £ 12 in the first year, rising annually to X30 or £35 as staff nurse, and subsequently to x;40 or £50 as sister or head nurse.

In the spectroscope calcium exhibits two intense lines-an orange line (a), (X 6163), a green line (a), (X 4229), and a fainter indigo line.

Potassium and sodium readily dissolve in the anhydrous acid with evolution of hydrogen and formation of x.

Let E be the effective elasticity of the aether; then E = pc t, where p is its density, and c the velocity of light which is 3 X 10 10 cm./sec. If = A cos" (t - x/c) is the linear vibration, the stress is E dE/dx; and the total energy, which is twice the kinetic energy Zp(d/dt) 2 dx, is 2pn2A2 per cm., which is thus equal to 1.8 ergs as above.

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.

We may now, as is somewhat the more natural course in the terrestrial application, take axes (x,y,z) which move with the matter; but the current must be invariably defined by the flux across surfaces fixed in space, so that we may say that relation (i) refers to a circuit fixed in space, while (ii) refers to one moving with the matter.

Trains of waves nearly but not quite homogeneous as regards wave-length will as usual be propagated as wave-groups travelling with the slightly different velocity d(VX-1)/dX-', the value of K occurring in V being a function of X determined by the law of optical dispersion of the medium.

Acts x.

Such in the main remained the situation until 1816, duties being indeed raised from x xv1.

The equations of motion are now, the co-ordinates x and y being measured in feet, 2 (26) - -rr- - C, dt2 dty - g' * These numbers are taken from a part omitted here of the abridged ballistic table.

Dividing again by x, as given in (76), tan0.

The field covers a district about 8 X 10 m.

We have such analytical formulae as that is, any number from I to 15 can be made with the parts I, 2, 4, 8; and similarly any can be made up, and in one way only, with A like formula is I - x 3 I - x9 I - x27 I x.

In the oceans alone there are estimated to be 1141 X 10 12 tons of sulphate, K 2 SO 4, but this inexhaustible store is not much drawn upon; and the "salt gardens" on the coast of France lost their industrial importance as potash-producers since the deposits at Stassfurt in Germany have come to be worked.

The problem of its earlier history is so entangled with the similar questions raised by x that the two cannot well be discussed separately.

This is a fairly strong case, but it falls short of demonstration because it cannot be shown that the MS. corrected by Pamphilus was still at Caesarea when it was used by x, and because it is not certain either that the chapter divisions in Acts were added by the original scribes, or that x and B were at that time in their original home, or that the chapter divisions were necessarily only to be found at Caesarea.

Up to 1816-1817 these societies had printed altogether 436,000 copies of the Scriptures, and had received from the British and Foreign Bible Society gifts amounting to over X 62,000.

That this unit is quite distinct from the Persian 86 grains is clear in the Egyptian weights, which maintain a wide gap between the two systems. Next, in Syria three inscribed weights of Antioch and Berytus (18) show a mina of about 16,400, or 200 x 82.

Pliny states the Egyptian talent at 80 librae = 396,000; evidently = the Abydus lion talent, which is divided by 100, and the mina is therefore 3960, or 50 x 79.2.

The largest weight is the "wood" talent of Syria (18) = 6 Roman talents, or 1,860,000, evidently 120 Antioch minae of 15,500 or 2 x 7750.

So we have thus a weight of 207-191 in Egypt on marked weights, joining therefore completely with the Aeginetan unit in Egypt of 199 to 186, and coinage of 199, and strongly connected with Syria, where a double mina of Sidon (18) is 10,460 or 50 x 209.2.

At Athens it was 2 x 4900, and on the average of all the Greek weights it is 2 x 4825, so that 4950 -- the libra -- is as close as we need expect.

A beautiful set of multiples of the scripulum was found near Lyons (38), from 1 to 10 x 17.28 grains, showing a libra of 4976.

Among them (29) the two contiguous groups can be discriminated by the 129 being multiplied by 30 and 60, while the 67 or 134 is differently x 25, 40, 50 and l00.

This relation between x, a, rn, may be expressed also by the equation x= log m.

For the purpose of thus simplifying the operations of arithmetic, the base is taken to be Io, and use is made of tables of logarithms in which the values of x, the logarithm, corresponding to values of m, the number, are tabulated.

It follows very simply from the definition of a logarithm that logo b X logo a, = 1, logo m =log.

The mathematical function log x or log x is one of the small group of transcendental functions, consisting only of the circular functions (direct and inverse) sin x, cos x, &c., arc sin x or sin-' x,&c., log x and e x which are universally treated in analysis as known functions.

The notation log x is generally employed in English and American works, but on the continent of Europe writers usually denote the function by lx or lg x.

The logarithmic function is most naturally introduced into analysis by the equation log x= x ?

This equation defines log x for positive values of x; if o the formula ceases to have any meaning.

Thus log x is the integral function of 1/x, and it can be shown that log x is a genuinely new transcendent, not expressible in finite terms by means of functions such as algebraical or circular functions.

A connexion with the circular functions, however, appears later when the definition of log x is extended to complex values of x.

The following fundamental properties of log x are readily deducible from the definition (i.) log xy= log x-Flog y.

Either of these properties might be taken as itself the definition of log x.

There is no series for log x proceeding either by ascending or descending powers of x, but there is an expansion for log (I +x), viz.

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

It has the important property that it tends to infinity with x, but more slowly than any power of x, i.e.

The exponential function possesses the properties (i.) exp (x+y) =exp x X exp y.

It is customary, therefore, to denote the exponential function by e x, and the result ex = I +x+x2/2 !

The definitions of the logarithmic and exponential functions may be extended to complex values of x.

Thus even when the argument is real log x has an infinite number of values; for putting 71 =o and taking positive, in which case a = o, we obtain for log the infinite system of values log +2n7ri.

It follows from this property of the function that we cannot have for log x a series which shall be convergent for all values of x, as is the case with sin x and cos x, for such a series could only represent a uniform function, and in fact the equation log(I +x) =x -",, x2 +3x 3 -4x 4 + is true only when the analytical modulus of x is less than unity.

The exponential function, which may still be defined as the inverse of the logarithmic function, is, on the other hand, a uniform function of x, and its fundamental properties may be stated in the same form as for real values of x.

An alternative method of developing the theory of the exponential function is to start from the definition exp x = I +x+x2/2 !

Logistic or Proportional Logarithms. - The old name for what are now called ratios or fractions are logistic numbers, so that a table of log (a/x) where x is the argument and a a constant is called a table of logistic or proportional logarithms; and since log (a/x) =log a-log x it is clear that the tabular results differ from those given in an ordinary table of logarithms only by the subtraction of a constant and a change of sign.

Corresponding to the argument log x it gives the values of log (I -Fx - 1) and log (1+x).

He then by means of a simple proportion deduced that log (I 00000 00000 00000 I)=o 00000 00000 00000 0 434 2 944 81 90325 1804, so that, a quantity 1.00000 00000 00000 x (where x consists of not more than seventeen figures) having been obtained by repeated extraction of the square root of a given number, the logarithm of I 00000 00000 00000 x could then be found by multiplying x by 00000 00000 00000 04342 To find the logarithm of 2, Briggs raised it to the tenth power, viz.

Since loge(I +x) =x-2x 2 -3x 3 - 4x4+&c., we have, by changing the sign of x, log e (I - x) _ - x - zx 2 - 3x 3 - x 4 - &c.; whence g 1 +x to=2(x+ix'+1x5+&c.), e l - x and, therefore, replacing x by p +q, log e q =2 p +q +3 () 3T ?

In the corresponding antilogarithmic process the number is expressed as a product of factors of the form 1+.i"x.

X 67rapSos) was given by the ancients to an animal supposed to have been a cross between a lion (Lat.

Real information as to the nations of Mexico before Spanish 1 In this, as in all other Aztec names, the x (or j) represents the English sound sh; hence Mexitli and Mexico should be properly pronounced Meshitli, Meshico.

But they do not appear to have ever been so pronounced by the Spaniards, who naturally gave to the x its ordinary Spanish sound of the German ch.

In the centre, the oblong pyramid of rubble cased with hewn stone and cemented 375 X 300 ft.

Seler, Verhandlungen des X VI.

Each " saying " is introduced by the phrase Jesus says " (XE'y€) and the collection is described in the introductory words of the 1903 series as Xoyyoc not as X&yca.

The sign of summation X is used in cases where there are several absorption bands, and consequently several similar terms on the right-hand side, each with a different value of A m.

A portion of Pierce's correspondence has been published in the American Historical Review, x.

The use of the words " after being filled," in x.

But Paul, while he saw this much in it, saw much more; or he could not in the same epistle, x.

The two "hospitia" or "guest-houses" for the entertainment of strangers of different degrees (X, X2) comprise a large common chamber or refectory in the centre, surrounded by sleeping-apartments.

Between it and the transept we find the sacristy (X), and a small book-room (Y), armariolum, where the brothers deposited the volumes borrowed from the library.

Obviously these equations show that the curves intersect in four points, two of which lie on the intersection of the line, 2 (g - g')x +2 (f - f')y+c - c'=o, the radical axis, with the circles, and the other two where the lines x2+y2= (x+iy) (x - iy) =o (where i = - - I) intersect the circles.

The equation x 2 +y 2 =o denotes a pair of perpendicular imaginary lines; it follows, therefore, that circles always intersect in two imaginary points at infinity along these lines, and since the terms x 2 +y 2 occur in the equation of every circle, it is seen that all circles pass through two fixed points at infinity.

Since the equation to a circle of zero radius is x 2 +y 2 =o, i.e.

The equations to such circles may be expressed in the form x 2 +y 2 = a 2, x 2 +y 2 = /3 2 .

These equations show that the circles touch where they intersect the lines x 2 +y 2 = o, i.e.

To construct circles coaxal with the two given circles, draw the tangent, say XR, from X, the point where the radical axis intersects the line of centres, to one of the given circles, and with centre X and radius XR describe a circle.

Analytically, the Cartesian equation to a coaxal system can be written in the form x 2 + y 2 + tax k 2 = o, where a varies from member to member, while k is a constant.

The radical axis is x = o, and it may be shown that the length of the tangent from a point (o, h) is h 2 k 2, i.e.

The value of the city's factory products increased from x$1,615,959 in 1900 to $3,146,890 in 1905, or 94-7%.

Internal pore leading from the non-glandular portion of the left nephridium to the external pore x.

The left inner gill-plate is also snipped to show the subjacent orifices of the left renal organ x, and of the genital gland (testis or ovary) y.

In the same figure the free part of the inner lamella of the inner gill-plate resting on the foot is marked z, whilst the attached parjt - the most anterior - has been snipped with scissors so as to show the genital and nephridial apertures x and y.

The readings of X which can be deduced from considering the agreements in B, C, D will be of higher antiquity and of greater external aut l ority than any of the readings in B, C, D taken singly.

Glover, The Conflict of Religions tin the Early Roman Empire, chap. x.

Moreover, by adding (Politics, H 7, 1327 b 29-33) that the Greek race could govern the world by obtaining one constitution (was Tvy X b.vov 7roXtmeias), he indicated some leaning to a universal monarchy under such a king as Alexander.

Lastly, pleasure, after having been first defined (Book vii.) as an activity, is treated over again (Book x.) as an end beyond activity, with a warning against confusing activity and pleasure.

Secondly, the Eudemian Ethics, while not agreeing with Plato's Republic that the just can be happy by justice alone, does not assign to the external goods of good fortune (Eutu X ia) the prominence accorded to them in the Nicomachean Ethics as the necessary conditions of all virtue, and the instruments of moral virtue.

It divides (chap. 8) evidences (7r1aTEts) into two kinds (I) evidence from arguments, actions and men (ai j s v E air&v Tcev Xhywv cal Twv 7rpit aw cal TWV avOpcoirwv); (2) adventitious evidences (ai S' iIriOETOtTOLs X yo,u vots cal Tois rpm-To/lb/0a).

Aristotle was the founder of Logic; because, though others, and especially Plato, had made occasional remarks about reason (X yos), Aristotle was the first to conceive it as a definite subject of investigation.

By Inserting, Therefore, Three Additional Months Instead Of Four In Every Period Of Eight Years, The Coincidence Between The Solar And Lunar Year Would Have Been Exactly Restored If The Latter Had Contained Only 354 Days, Inasmuch As The Period Contains 354X8 3 X 30 = 2922 Days, Corresponding With Eight Solar Years Of 3654 Days Each.

On The Other Hand, The Exact Time Of A Synodic Revolution Of The Moon Is 29'530588 Days; 235 Lunations, Therefore, Contain 2 35 X 29 530588 = 6 939'6 8818 Days, Or 6 939 Days 16 Hours 31 Minutes, So That The Period Exceeds 235 Lunations By Only Seven And A Half Hours.

The Number Of Years In The Intercalary Period Being Four, And The Days Of The Week Being Seven, Their Product Is 4 X 7 = 28; Twenty Eight Years Is Therefore A Period Which Includes All The Possible Combinations Of The Days Of The Week With The Commencement Of The Year.

This Rule Is Conveniently Expressed By The Formula (X 9) R, In Which X Denotes The Date, And The Symbol R Denotes That The Remainder, Which Arises From The Division Of X 9 By 28, Is The Number Required.

The Symbol X, Employed In The Formula At The Top Of The Column, Denotes The Number Of Centuries, That Is, The Figures Remaining After The Last Two Have Been Struck Off.

This gives 19 X354+6 X 30429 = 6 935 days, to be distributed among 235 lunar months.

The cycle of the sun brings back the days of the month to the same day of the week; the lunar cycle restores the new moons to the same day of the month; therefore 28 X 19 = 53 2 years, includes all the variations in respect of the new moons and the dominical letters, and is consequently a period after which the new moons again occur on the same day of the month and the same day of the week.

This number is called the Golden Number, either because will therefore be given by the formula (x 1 53) r, that is to say, add 3 totlzedate,divide the sum by 15, and the remainder is the year of the indiction.

The Julian period, proposed by the celebrated Joseph Scaliger as an universal measure of chronology, is formed by taking the continued product of the three cycles of the sun, of the moon, and of the indiction,and is consequently 28 X 19X I 5= 7980 years.

Let X, Y, And Z Be The Three Quotients Of The Divisions; The Number Sought Will Then Be Expressed By 28 X Io, By 19 Y 2, Or By 15 Z 4.

To Solve The Equations 28 X Io =19 Y 2, Or Y =X 9198, Let M= 9 19 8, We Have Thenx=2 M 172 8 9 Let M9 8 =M'; Then M= 9 M' 8; Hence X=18 M' 16 M'=19 M' 16 (I).

Equating The Above Two Values Of X, We Have 15 N' 3 =19 M' 16; Whence N' =M' 4'N1 13.

X, Period, 47 1 4, 47 1 5, 4716,...4713 X; From Which It Is Evident, That If We Take P To Represent The Year Of The Julian Period, And X The Corresponding Year Of The Christian Era, We Shall Have P= 47 1 3 X, And X=P 4713.

In The Year Preceding The First Of The Era, The Dominical Letter Was C; For That Year, Therefore, We Have L =3; Consequently For Any Succeeding Year X, L =7M 3 X, The Years Being All Supposed To Consist Of 365 Days.

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.

After X Centuries, Therefore, It Must Be Diminished By X (4).

This Gives Eight To Be Added In A Period Of Twenty Five Centuries, And 85 In X Centuries.

P. Pfeiffer (Berichte, 1904, 37, p. 4 2 55) has shown that chromium salts of the type [Cr{C2H4(NH2)2}2X2]X exist in two stereo-isomeric forms, namely, the cisand transforms, the dithiocyan-diethylenediamine-chromium salts being the transsalts.

Let somehow or other retardations be introduced so that the optical length of the successive parts increases by the same quantity nX, n being some number and X the wave-length.

B, a specimen undergoing fission (X 20).