A b X N N(A) N+2N(A)+A No other rational explanation of the close relationship between albinism and cousin marriages is at present forthcoming.
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.
A connexion with the circular functions, however, appears later when the definition of log x is extended to complex values of x.
A knowledge of the absolute value of this energy is not required, since in every expression in which it occurs it is under the (3) form x - xo, that is to say, the difference between the energy in two different states.
A list of them, with notice of Anselm's life, is contained in the Histoire litteraire de la France, x.
A machine of this kind, if it printed a sheet of double demy, which measures 35 X 221 in., was about 13 ft.
A notice of its history in 147 B.C. is found in 1 Macc. x.
A portion of Pierce's correspondence has been published in the American Historical Review, x.
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.
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.
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.
Abteilung, pp. 216-226 and 440-442; Ludwig Braunfels, Kritischer Versuch fiber den Roman Amadis von Gallien (Leipzig, 1876); Theophilo Braga, Historia das novelas portuguezas de cavalleria (Porto, 1873), Curso de litteratura e arte portugueza (Lisboa, 1881), and Questoes de litteratura e arte portugueza (Lisboa,1885); Marcelino Menendez y Pelayo, Origenes de la novela (Madrid, 1905); Eugene Baret, De l'Amadis de Gaule et de son influence sur les me urs et la litterature au X VI e et au X VII e siecle (Paris, 1873).
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).
Addai was supposed to be one of the Seventy of Luke x.
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.
After X Centuries, Therefore, It Must Be Diminished By X (4).
Again the Amphictyonic unions had one of the x.
Again, if x, y, z be the co-ordinates of P, the component velocities of m are qzry, rxpz, pyqx, (6)
Also, if 0 be the angle between them, and x", y", z" the direction-cosines of a line perpendicular to each of them, we have xx' +yy'+zz' =cos 0, yz' - zy" = x" sin 0, &c., so that the product of two unit lines is now expressed as - cos0+ (ix" +jy" +kz") sin 0.
Although his data were the proper motions of only seven stars, he indicated a point near X Herculis not very far from that found by modern researches.
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.
An alternative method of developing the theory of the exponential function is to start from the definition exp x = I +x+x2/2 !
An important property of the diagram is that if points X and x are taken dividing the link CB and the whole acceleration of B about C, namely, cb in the same ratio, then Ax represents the acceleration of the point X in magnitude and direction; cb is called the acceleration image of the rod.
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.
Arabic and other writers are given in Ritter, Erdkunde x.
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.
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.
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.
As an historian he published Origines de l'institution des intendants de provinces (1884), which is the authoritative study on the intendants; Etudes historiques sur les X VI e et X VIP siecles en France (1886); Histoire de Richelieu (2 vols., 1888); and Histoire de la Troisieme Republique (1904, &c.), the standard history of contemporary France.
As an instance leading to results of some importance consider the series x x2 F(n,x) =I+ (y+n)I!+(y+n)(y+n+I)2!+..
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.).
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.
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.
As x is found in the same inscription (in the form X), the guttural element must have been different, else would have been spelt x*.
Assuming that r=a+x, where x is small, we have, approximately, dix hiI 3x\
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.
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.
At Charles X.'s death, five years later, we find Sweden, herself bled to exhaustion point, surrounded by a broad belt of desolated territory and regarded with ineradicable hatred by every adjacent state.
At points in the liquid very near its surface it is probable that x is greater than xo, and at points in the gas very near the surface of the liquid it is probable that x is less than x', but this has not as yet been ascertained experimentally.
Attica and most of the Cyclades kept x for the guttural element in (written x5 or + 5) and for X as well.
B where x = the distance between the two lines = N /No.
B, a specimen undergoing fission (X 20).
B, Archigetes sieboldi; X 60 (from Leuckart).
B, head showing the suckers, proboscides and excretory canals; X 25.
Becquerel's observation in 1896 that certain uranium preparations emitted a radiation resembling the X rays observed by Rntgen in 1895.
Benoist's Guichardin, historien et homme d'etat italien an X VI' siecle (Paris, 1862), and C. Gioda's Francesco Guicciardini e le sue opere inedite (Bologna, 1880) are not without value, but the authors had not had access to many important documents since published.
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.
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 .
Both narratives are doubtless based upon actual occurrences - the cures narrated in Mark ii., iii., viii., x.
Brussels was again the headquarters in 1741, by which time Voltaire had finished the best and the second X XVIII.
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.
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.
But Paul, while he saw this much in it, saw much more; or he could not in the same epistle, x.
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.
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.
But, if no solid be present initially, or if the cooling be rapid, the liquid of composition x becomes supersaturated and may cool till the supersaturation curve is reached at b, and a cloud of A crystals comes down.
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.
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.
By putting =x 2 /4 for x in F(o,x) and F(i,x), and putting at the same time y =1/2, we obtain x 2 x 2 x2 x 2 x2 tan x x x tanh x x x x I - 3 - 5-7-...
C, invaginated head of Cysticercus cellulosae, showing the bent neck and receptacle r; X 30.
C. Rntgen of Munich made in 1896 his remarkable discovery of the so-called X or Rntgen rays, a class of radiation produced by the impact of the cathode particles against an impervious metallic screen or anticathode placed in the vacuum tube.
Chinchillone.-Size 13 X 8 in., obtained also from South America.
Cj = (Moga/c)y+aQ, C4 = (Moga/c)x aP, ct = 0.
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.
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.
Corresponding to the argument log x it gives the values of log (I -Fx - 1) and log (1+x).
Dividing again by x, as given in (76), tan0.
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.
Each of the above formulae involves an arbitrary constant; but this disappears when we start the additions from 'a known value of X udx.
Each of the four co-ordinates, n', x', y are functions of %, rj, x, y; and if it be assumed that the field of view and the aperture be infinitely small, then, n, x, y are of the same order of infinitesimals; consequently by expanding ', ii', x', y in ascending powers of E, rt, x, y, series are obtained in which it is only necessary to consider the lowest powers.
Either of these properties might be taken as itself the definition of log x.
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.
English, though possessing sounds comparable to the Greek 0, /, x, has never made any attempt to represent them in writing.
Equating The Above Two Values Of X, We Have 15 N' 3 =19 M' 16; Whence N' =M' 4'N1 13.
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.
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.
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.
For example, in a one-dimensional system such as a string or a bar, we have one dependent variable, and two independent variables x and t.
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).
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.
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.
For this reason X is also known as the angle of repose.
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...
From the subjectivist point of view, which is manifestly fundamental through most of this, such arguments suasory of the Pyrrhonist suspense of judgment (i ro X i i) are indeed hard to answer.
From this point of view the equation is a mere truism, its real importance resting on the fact that by attributing suitable values to the masses in, and by making simple assumptions as to the value of X in each case, we are able to frame adequate representations of whole classes of phenomena as they actually occur.
Generally the larger of these machines will print a sheet up to 21 X 16 in.
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.
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.
Glover, The Conflict of Religions tin the Early Roman Empire, chap. x.
Grignard (Comptes rendus, 1900 et seq.) observed that magnesium and alkyl or aryl halides combined together in presence of anhydrous ether at ordinary R temperatures (with the appearance of brisk boiling) to form compounds of the type RMgX(R = an alkyl or aryl group and X = halogen).
H = wL2/8y, or, calling x the distance from the vertex to the point of support, H = wx2/2y.
Hamilton, still keeping prominently before him as his great object the invention of a method applicable to space of three dimensions, proceeded to study the properties of triplets of the form x+iy+jz, by which he proposed to represent the directed line in space whose projections on the co-ordinate axes are x, y, z.
He died on the 1st of January 1515 For a bibliograply of the printed sources see Henri Hauser, Les Sources de l'histoire de France, X VI e siecle, vol.
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.
He thinks that the guttural element in E was a spirant, and therefore different from X, which is an aspirate.
He took as his secretary of state Cardinal Raphael p us x Merry del Val, a Spaniard of English birth and educa tion, well versed in diplomacy, but of well-known ultramontane tendencies.
He was not a Sadducee, for he looks forward to the establishment of the Messianic Kingdom (x.).
He was then director of the college of Senlis, where he composed his Esprit de la Ligue ou histoire politique des troubles de la Fronde pendant le X VI e et le X Vil e siecles (1767).
He wrote a X povoypa4La in 18 books, the beginning and the end of which are lost.
Hence if 1 be the length of the string, and x the horizontal displacement of the bob from the equilibrium position, the horizontal component of gravity is mgx/l, whence The motion is therefore simple-harmonic, of period r= 27ri/(l/g).
Hence if we write T= f E (x - xo) p d v,.
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.
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).
Hence the forces x, y, x, y are in equilibrium.
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.
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.
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.
Here, then, is a case specially adapted to the isotropy of the quaternion system; and Hamilton easily saw that the expression i d x +j - + k dz could be, like ix+jy+ kz, effectively expressed by a single letter.
Heusinger has shown that white sheep and pigs are injured by the ingestion (A) N b X N ?--,r N(A)+N of certain plants, while the pigmented individuals may eat them without harm.
His temple was about 40 X 50 ft.
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.
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.
If 01 be drawn vertically upwards, then in the beginning of the disturbed motion the quantities x, y, p, q, P, Q will all be small.
If any other point 0, whose co-ordinates are x, y, z, be chosen in place of 0, as the point to which the forces are transferred, we have to write x1x, yiy, ZiZ for xl, Yi, z1,and so on, in the preceding process.
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.
If dw/dt is the angular acceleration of the link, dw/dt X CB is the tangential acceleration of the point B about the point C. Generally this tangential acceleration is unknown in magnitude, and it becomes part of the problem to find it.
If in (21) we imagine that x, y, I denote infinitesimal rotations of a solid free to turn about a fixed point in a given field of force, it appears that the three normal modes consist each of a rotation about one of the three diameters aforesaid, and that the values of in are proportional to the ratios of the lengths of corresponding diameters of the two quadrics.
If L is zero or negligible, X = - P(v - v) and we have dP/dT = - P/T or dP/P = - dT/T, which on integration gives log P =log T+C, or P= kT, i.e.
If n is even, solutions of the equation x 2 -Ny 2 =+1 can be found.
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).
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.
If Q= Ep+nq+wr and we put Q= (I +Zwt)(Ep-i-nq)X (1 Zwt) -1 we find that the quaternion t must be 2f (r) /f (q - p), where f(r)=rq - Kpr.
If the axis of x be taken parallel to the slope of the plane, with x increasing downwards, we have -
If the capacity C is reckoned in microfarads then the energy storage is equal to CV 2 /2 X 19 6 joules or 0.737 CV 2 / 2 X 10 6 foot-pounds.
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'?
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.
If the extraneous forces be re p duced to a force (X, Y) at G and a couple N, we have Mx=X,My=Y,IO=N.
If the force, X be always the same in the same position, the particle may be regarded as moving in a certain invariable field of force.
If the liquid forms a stratum of thickness c, then x = x' - 47rpo(o) +27rpo(z) +27rpo(Z - c).
If the liquid is bounded by a dense substance, whether liquid or solid, the value of x may be different from its value when the liquid has a free surface.
If the load rests directly on the main girder, the greatest -Iand - shears at C will be wXAGC and -w X CHB.
If the origin of rectangular axes fixed in the lamina be shifted through a space whose projections on the original directions of the axes are X, u, and if the axes are simultaneously turned through an angle e, the coordinates of a point of the lamina, relative to the original axes, are changed from x, y to X+x cos ey sin e, u+x sin e+y cos e, or X + x ye, u + Xe + y, ultimately.
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.
If the thickness of the film is greater than 2E, there will be a stratum of thickness c-2E in the middle of the film, within which the values of p and x will be pc and In the two strata on either side of this the law, according to which p and x depend on the depth, will be the same as in a liquid mass of large dimensions.
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 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 the wave-length is X, the equation of the surface is y=b sin 2lrxx The pressure due to the surface tension T is p= - Td 2 = 4Ty.
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.
If then y=a sin (x - Ut),, t=2,ro-ra cos (x - Ut).
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 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!
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.
If two roots are imaginary the equation is y 2 =(x 2 +a 2) (x - b) and the curve resembles the parabolic branch, as in the preceding case.
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+..
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.
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 We Assume M A Tx.E.Vz.A", We Have, Considering In Turn Length, Time And Mass, Y 3Z U=O, 2X 2Y=O, X Z=I; So That Y= X, Z =1 X, U=3 2X.
If we now apply them to the case of a rigid body moving about a fixed point 0, and make Ox, Oy, Oz coincide with the principal axes of inertia at 0, we have X, u, v=Ap, Bq, Cr, whence A (B C) qr = L,
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 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.
If we take as rough values a=21 X,o6 feet, g=32 foot-second units, we get a velocity of 36,500 feet, or about seven miles, per second.
If we take the axis of z normal to either surface of the film, the radius of curvature of which we suppose to be very great compared with its thickness c, and if p is the density, and x the energy of unit of mass at depth z, then o- = f o dz, (16) and e = f a xpdz,.
If we transfer 0 to A, e=o, and the curve may be represented by y=a sin A x.
If X ---- r - i the scales are so chosen i that a inch represents 1 x ' n - "--, in.
If x be measured upwards from the lower end, the horizontal component of the tension P at any point will be Pay/ax, approximately, if y denote the lateral displacement.
If x denotes the potential energy of unit of mass of the substance, we may treat x as sensibly constant except within a distance e of the bounding surface of the fluid.
If x' is the potential energy of unit of mass of the substance in vapour, then at a distance z from the plane surface of the liquid X = X' - 22 7rp 7rpe e ((zo)) ..
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.
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.
In 1686, fearing that they might be dispossessed by a new charter, the people of Salem for X 20 secured a deed from the Indians to the land they then held.
In addition to these marks, an X may appear before, after or in both places.
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.
In connexion with the journey from this region to Jerusalem three striking incidents are recorded, x.
In its modern form the theorem, which is true for all values of n, is written as (x +a) n -1+ I.
In most judgments all we believe is that x is (or is not) y, that a thing is (or is not) determined, and that the thing signified by the subject is a thing signified by the predicate, but not that it is the only thing, or equal to everything signified by the predicate.
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 the a x = aixi+a2x2, observe that a a = a2, ab = aibi +a2b2.
In the case of a constant force, the acceleration or x is, according to (1), constant, and we have say, the general solution of which is x=1/8ati+At+B.
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.
In the case of a recurring continued fraction which represents N, where N is an integer, if n is the number of partial quotients in the recurring cycle, and pnr/gnr the nr th convergent, then p 2 nr - Ng2nr = (- I) nr, whence, if n is odd, integral solutions of the indeterminate equation x 2 - Ny 2 = I (the so-called Pellian equation) can be found.
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.
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 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, 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.
In the centre, the oblong pyramid of rubble cased with hewn stone and cemented 375 X 300 ft.
In the Chalcidian alphabet the symbol for x was placed after the symbols common to all Greek alphabets, a position which X retains in the Latin (and also in the Faliscan) alphabet.
In the corresponding antilogarithmic process the number is expressed as a product of factors of the form 1+.i"x.
In the equation dP/dT= X/T(v 2 - v 1), P is the osmotic pressure, T the absolute temperature and X the heat of solution of unit mass of the solute when dissolving to form a volume v2 - v1 of saturated solution in an osmotic cylinder.
In the equations of uniplanar motion = dx - du = dx + dy = -v 2 ?, suppose, so that in steady motion dx I +v24 ' x = ?'
In the geometry of plane curves, the term parabola is often used to denote the curves given by the general equation a' n x n = ym+n, thus ax= y 2 is the quadratic or Apollonian parabola; a 2 x = y 3 is the cubic parabola, a 3 x = y4 is the biquadratic parabola; semi parabolas have the general equation ax n-1 = yn, thus ax e = y 3 is the semicubical parabola and ax 3 = y 4 the semibiquadratic parabola.
In the interior of the fluid it has the uniform value X i.
In the momentum equation (4) we may now omit X and it becomes 0.+P(U - u) 2 =poU2.
In the more "extreme" x 5a churches the surplices are frank imitations of the Roman cotta.
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 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.
In the remaining paragraph, x.
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.
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.
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.
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.
In the steady state, the product kdO/dx must be constant, or the gradient must vary inversely as the conductivity, if the latter is a function of 0 or x.
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.
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.
In this case the solution of the equation reduces to the form e =x(1 - x)C 2 Ro/2lgk.
In this simple case the temperature cycle at a depth x is a precisely similar curve of the same period, but with the amplitude reduced in the proportion rn ', and the phase retarded by the fraction mx/27r of a cycle.
Internal pore leading from the non-glandular portion of the left nephridium to the external pore x.
It appears that the rapidity with which a x rubbing surface wears away is proportional to Fio.
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.).
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."
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).
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.
It follows very simply from the definition of a logarithm that logo b X logo a, = 1, logo m =log.
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.
It has the important property that it tends to infinity with x, but more slowly than any power of x, i.e.
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.
It is customary, therefore, to denote the exponential function by e x, and the result ex = I +x+x2/2 !
It is implied in the above description of the system that the Cartesian co-ordinates x, y, z of any particle of the system are known functions of the qs, varying in form (of course) from particle to particle.
It is necessary that the thickness x should be approximately uniform.
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.).
It is obvious that the ratio V (x,y,z) (22)
It is practically only bulls of canonization which are signed by the pope and all the cardinals present in Rome; the signature of the pope is then "(Pius) Episcopus Ecclesiae catholicae," while his ordinary signature bears only his name and number, "Pius PP. X."
It is seen that for a given direction of p this moment is least when p passes through the intersection X of the first and last sides of the funicular; i.e.
It may be noticed that if the scales of x and be properly adjusted, the curve of positions in the present problem is the portion of a cycloid extending from a vertex to a cusp.
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.
Its angular points have then co-ordinates (x t Zdx, y t Zdy, z * zdz).
Its cartesian equation is x 3 -1-y 3 =3axy.
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.
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.
Let a be the radius of the rolling sphere, c that of the spherical surface which is the locus of its centre, and let x, y, I be the co-ordinates of this centre relative to axes through 0, the centre of the fixed sphere.
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.
Let KA and LB be the positions of NP corresponding to the extreme values of x.
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.
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- .
Let the surface of separation be originally in the plane of the orifice, and let the co-ordinates x and y be measured from one corner parallel to the sides a and b respectively, and let z be measured upwards.
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).
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.
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.
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.
Like the X rays, the Becquerel rays are invisible; they both traverse thin sheets of glass or metal, and cannot be refracted; moreover, they both ionize gases, i.e.
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.
Maitland describes it (Political Theories of the Middle Ages, p. x.), have an essentially "statelike character."
Many of these machines are made to print four double crowns, 60 X 40 in., or even larger.
Mary and Martha are admittedly identical with the sisters in Luke 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.
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.
No terms in x or x 2 need be considered.
Nor yet was he a Zealot, for the quietistic ideal is upheld (ix.), and the kingdom is established by God Himself (x.).
Note that these phagocytic cells are pushing out protoplasmic processes (pseudopodia) by which they grasp their victims. (X woo diam.) FIG.
Now consider a portion of a cylindric film of length x terminated by two equal disks of radius r and containing a certain volume of air.
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.
Now it is shown in hydrodynamics that she velocity of propagation of waves in deep water is that acquired by a heavy body falling through half the radius of the circle whose circumference is the wave-length, or _ f_X _ ga 27rT 'I ' v2- 2r 2r pn This velocity is a minimum when X=2.7r gp' and the minimum value is v= 4 - p g For waves whose length from crest to crest is greater than X, the principal force concerned in the motion is that of gravitation.
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.
Now x, x have a resultant through H, represented in magnitude and direction by 00, whilst y,y have a resultant through K represented in magnitude and direction by OO.
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.
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.
On account of the aberrations of all rays which pass through 0, a patch of light, depending in size on the lowest powers of E, x, y which the aberrations contain, will be formed in the plane I'.
On Charles X.'s accession in 1655, Sweden's neighbours, though suspicious and uneasy, were at least not adversaries, and might have been converted into allies of the new great power who, if she had mulcted them of territory, had, anyhow, compensated them for the loss with the by no means contemptible douceur of religious liberty.
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.
On the hypothesis of uniform density we shall find that this is true for films whose thickness exceeds The symbol x is defined as the energy of unit of mass of the substance.
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.
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.
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.
Others also before Menes are 15 X 25 ft.
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?
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.
P2 = x2 I A I + x 2 I A I +.
Plato (Rep. x.) embodies the idea in one of his finest myths.
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.
Politics in Finland were complicated by the rivalry between the Swedish party, which x.
Potassium and sodium readily dissolve in the anhydrous acid with evolution of hydrogen and formation of x.
Proceeding to the general motion of a rigid body in two dimensions we may take as the three co-ordinates of the body the rectangular Cartesian co-ordinates x, y of the mass-centre G and the angle C through which the body has turned from some standard position.
Putting A /M =n 2 the equation becomes x+n 2 x=o, whence x =A sin nt, and the period is 27r/n.
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..
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.
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.
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.
Seler, Verhandlungen des X VI.
Similarly the corresponding image ray may be defined by the points (t', i'), and (x', y'), in the planes I' and II'.
Similarly, rs = W (x +a) 11 is the shear on DB.
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,, ?
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 ?
Since R2=X2+Y2+Z2, G2=L1+M2+N2, it is necessary and sufficient for equilibrium that the six quantities X, Y, Z, L, M, N, should all vanish.
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 ?
Since the equation to a circle of zero radius is x 2 +y 2 =o, i.e.
Since the projection of a vector- sum is the sum of the projections of the several vectors, the equation (2) gives if x be the projection of 0G.
Since the two sides of the stratum are similar the last two terms are equal, and Differentiating with respect to c, we find dc = P' d p(x ' -41rpe (0))+47rp20(c).
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.
Since y must vanish at the upper end (x 1), the admissible values of a are determined by o2=gz2/41, Jo(z)=o.
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.
Such curves are given by the equation x 2 - y 2 = ax 4 -1bx 2 y 2 +cy 4 .
Such in the main remained the situation until 1816, duties being indeed raised from x xv1.
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.
Symbols exactly like k, X, and (a), X, are found in the Carian alphabet, and transliterated by Professor Sayce 1 as v (and ii), h and kh respectively.
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.
T appears in Etruscan as y, 7 t, and X; of these Umbrian borrows the first two, while Oscan has a form T like Latin.
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.
Take any base X'X, and draw lines at right angles to this base through all the angular points of the figure.
Taking 7 = 1, writing x/a for x and increasing a indefinitely, we I x x x x x have e x = - 1-1+2-3+2-5+...
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.
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.
That the Germans were familiar with some sort of marks on wood at a much earlier period is shown by Tacitus's Germania, chap. x.
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.
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).
The amplitudes of these tones are proportional to the products of a and b multiplied by X or µ.
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.
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 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.
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."
The axis of the member xQ+x'Q' of the second-order complex Q, Q' (where Q=nq+wr, Q'=nq'+wr' and x, x' are scalars) is parallel to a fixed plane and intersects a fixed transversal, viz.
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 Book of Noah is mentioned in Jubilees x.
The canvas, 36 X 28 in., admitted of less than a half-length portrait but was sufficiently long to include a hand, and this is known as the kit-cat size.
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 '.
The central nervous system (x) is highly developed, and in Loxosoma bears a pair o` eyes.
The city occupies an area of about 21 X 12 sq.
The component velocities of any point whose co-ordinates relative to 0 are x, y, z are then u+qzry, v+rxpz, w+Pyqx (12)
The components of angular momentum about Ox, Qy, 01 will be denoted as usual by X, u, v.
The components of force, X, Y, and N, acting on the liquid at 0, and reacting on the body, are then X=It.
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.
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.
The couple M may be produced by shifting the force F parallel to itself through a distance x, such that Fx=M.
The curve is symmetrical about the line x = y, and consists of two infinite branches asymptotic to the line x+y+a = o and a loop in the first quadrant.
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.
The definitions of the logarithmic and exponential functions may be extended to complex values of x.
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.
The diffusivity can be deduced from observations at different depths x' and x", by observing the ratio of the amplitudes, which is (x '- x ") for a simple-harmonic wave.
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.
The earliest is about I X 20 ft.
The earliest of the groups of which the book is composed seems to be x.
The episode is known in American history as the " X Y Z Affair."
The equation of the latter, referred to its principal axes, being as in II (41), the co-ordinates of the point J where it is met by the instantaneous axis are proportional to p, q, r, and the direction-cosines of the normal at J are therefore proportional to Ap, Bq, Cr, or X, u, v.
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.
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.
The equations to such circles may be expressed in the form x 2 +y 2 = a 2, x 2 +y 2 = /3 2 .
The exponential function possesses the properties (i.) exp (x+y) =exp x X exp y.
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.
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 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 field covers a district about 8 X 10 m.
The finest of these apartments, containing beautiful arabesque x XVII.
The first section, doubtless, belongs to the main narrative of P; it connects directly with chap. x.
The first three terms express the work done by the components of a force (X, Y, Z) acting at 0, and the remaining three terms express the work of a couple (L, M, N).
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.
The following fundamental properties of log x are readily deducible from the definition (i.) log xy= log x-Flog y.
The force corresponding to AB may be replaced by the two components marked x, y; and a force corresponding to BA may be represented by the two components marked x, y.
The formula for length of scale is, length = sighting radius X tangent of the angle of elevation.
The formulae can be adapted to the case in which cp(x) is tabulated for xl,.
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.
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,
The function log x as x increases from o towards w steadily increases from - co towards +co.
The general continued fraction al is evidently equal, convergent by convergent, to the continued fraction X 2 b 2 X2X3b3 x3%4b4 a1+ A2a2 + X + X
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.
The greatest moment CG at C is x(l-x)/l.
The gross rental at first payable to the company was £53,000, but within a period of ten years it had risen to X 146,000.
The heat transmitted through the plane x is equal per unit area of surface to the product of k by the mean temperature gradient (de /dx) and the interval of time, T - T'.
The history of the symbols 4 and x is altogether unknown.
The hyperbolic or Gudermannian amplitude of the quantity x is ta n (sinh x).
The indignation excited by Leo X.'s sale of indulgences, the moral rage stirred in Northern hearts by papal abominations in Rome, were external causes which precipitated the schism between Teutonic and Latin Christianity.
The integration can be performed when M is expressed in terms of x.
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.
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.
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.
The Legion of Honour, the only order of France, and one which in its higher grades ranks in estimation with the highest European orders, was instituted by Napoleon Bonaparte on the 19th of May 1802 (29 Floreal of the year X.) as a general military and civil order of merit.
The Libel of English Policie, a poem of the first half of the 15th century, says with reference to Iceland (chap. x.) "Out of Bristowe, and costes many one, Men haue practised by nedle and by stone Thider wardes within a litle while."
The load on an element bx of the beam may be represented by wbx, where - w is in general a function of x.
The logarithmic function is most naturally introduced into analysis by the equation log x= x ?
The magnifying power of the telescope is = Ff /ex, where F and f are respectively the focal lengths of the large and the small mirror, e the focal length of the eye-piece, and x the distance between the principal foci of the two mirrors (=Ff in the diagram) when the instrument is in adjustment for viewing distant objects.
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 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.
The meaning of these quantities is easily recognized; thus X is the angular momentum about a horizontal axis normal to the plane of 0, u is the angular momentum about the vertical OZ, and s is the angular momentum about the axis of symmetry..
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).
The nature of the reproduction consists in the rays proceeding from a point 0 being united in another point 0'; in general, this will not be the case, for E', vary if, 77 be constant, but x, y variable.
The next case is that in which u is a quadratic function of x, i.e.
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 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.
The personality of Lazarus in John's account, his relation to Martha and Mary, and the possibility that John reconstructed the story by the aid of inferences from the story of the supper in Luke x.
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.
The position of the central ordinate is given by x = v 1 /po, and therefore is given approximately by pl/po.
The precise value of X will vary with the nature and condition of the surfaces in contact.
The problem is identical with that of finding the common conjugate diameters of the ellipsoids T(x, y, I) =const., V(x, y, 1) =const.
The problem of its earlier history is so entangled with the similar questions raised by x that the two cannot well be discussed separately.
The quantity of heat absorbed by the stratum (x' x") in the interval considered can also be expressed in terms of the calorimetric conductivity k.
The question is one x, Stout axile strand.
The radial distance x is at any instant proportional to the force acting through the spring.
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 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.
The relation between x and t in any particular case may be illustrated by means of a curve constructed with I as abscissa and x as ordinate.
The religious element is prominent in x.
The result was an amplitude of 1.27 X 107 cm.
The salts of scandium are all colourless, the chloride and bromide corresponding in composition to Sc 2 X 6.12H 2 0; the fluoride is anhydrous.
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.
The sign x was kept in the western group for the guttural spirant in E, which was written X*; but, as this spirant occurred nowhere else, the combination was often abbreviated, and X was used for X precisely as in the Italic alphabets we shall find that F =f develops out of a combination FH.
The simplest case is that in which u is constant or is a linear function of x, i.e.
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).
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.
The story of Jephthah's fight with Ammon is linked to the preceding introduction by x.
The structure of the epidermis of the under side of the leaf, with its contorted cells, is represented (X 160) in fig.
The surface-density of this stratum is a = cp. The energy per unit of area is e = f xpdz=cp(X' -4lrpe(o))+27rp'f c 0(z) dz+27rp fee(c - z)dz.
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.
The symbol x expresses the energy of unit of mass of the liquid at a depth v within the bounding surface.
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.
The temples successively built here on one site were nine or ten in number, from the Ist dynasty, 5500 B.C. to the XXVIth dynasty, Soo B.C. The first was an enclosure, about 30 X 50 ft., surrounded by a thin wall of unbaked bricks.
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.
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.
The two courts are divided by the main buildings of the monastery, including the church, the sanctuary (A), divided from B, the monks' choir, by a screen with two altars, the smaller cloister to the south (S) surrounded by the chapter-house (E), the refectory (X) - these buildings occupying their normal position - and the chapel of Pontgibaud (K).
The use of the words " after being filled," in x.
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..
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.
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%.
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 warp is commonly from 36 8 to 44 8, the weft from 36 8 to 54 8, and the threads from 13 X 13 to 20X 20 to the 4 in.
The wave at a depth x is represented analytically by the equation 0 - 0 0 = Ae mx sin (21rnt - mx).
The whole amounted to X 2,340,000.
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.
The words ascribed to Christ in Luke x.
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.
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.
Then we may, ignoring the units G and H, speak of ON and NP as being equal to x and u respectively.
Then, if the outside cylinder is free to move - 2a2 2 X 1 = 0, T T = b2 a2, 7rpa 2 Ub 2+ a 2.
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.
Then, while T is slid along the axis from 0 towards X, P will be drawn after it from R towards C along the tractory.
There are extant 6 Anecdota Oxoniensia, Classical Series, part x.
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.
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.
These can again be combined into a single resultant R acting in the direction (X, u, v), provided X=RX,YRu,ZRv.
These equations show that the circles touch where they intersect the lines x 2 +y 2 = o, i.e.
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.
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.
These variables represent the whole assemblage of generalized co-ordinates qr; they are continuous functions of the independent variables x, y, 1 whose range of variation corresponds to that of the index r, and of 1.
These vesicles are filled with the colloid material (x 90 diam.) J FIG.
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.
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.
Thirdly, the vials source from the time of Pompey (circa 63),- x.
This alliance between the two monarchs led to a 1 In Judges x.
This equation defines log x for positive values of x; if o the formula ceases to have any meaning.
This functionary is first formally mentioned under Leo X.(1513-1521) in the proceedings in connexion with the canonization of St Lorenzo Giustiniani.
This gives 19 X354+6 X 30429 = 6 935 days, to be distributed among 235 lunar months.
This Gives Eight To Be Added In A Period Of Twenty Five Centuries, And 85 In X Centuries.
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.
This leads to a determinantal equation in X whose 2n roots are either real and negative, or complex with negative real parts, on the present hypothesis that the functions T, V, F are all essentially positive.
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.
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.
This ordinate will be an algebraical function of x, and we can again apply a suitable formula.
This refers to the use of the x, y, z co-ordinates, - associated, of course, with i, j, k.
This relation between x, a, rn, may be expressed also by the equation x= log m.
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.
This shows that f E (x - xo) p dv must be negative for water in contact with glass.
This statement has given rise to considerable discussion, but from it and internal evidence the date of the Apology (IIpar/3Eia crepe X pl- a-rcavwv) may be fixed at about A.D.
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.
Thus a push or a compression of the X air is transmitted onwards in the direction OX.
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.
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.
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 =..
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.
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.
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.
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.
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.
Thus, we must think in (r) All P is M " to avoid illicit process of the major, in (2) "All y is z " to avoid undistributed middle, in (3) "All x is y" to avoid illicit process of the minor.
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.
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.
To find the small oscillation about a state of steady precession in which the axis makes a constant angle a with the vertical, we write O=a+X, and neglect terms of the second order in x.
To illustrate the importance of the mensuration of graphs, suppose that we require the average value of u with regard to x.
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.
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).
To take a second case, molten hypo-eutectoid steel of 0.20% of carbon on freezing from K to x passes in the like manner to the state of solid austenite, -y-iron with this 0.20% of carbon dissolved in it.
To take the simple case of the " wall " or flat plate considered by Fourier for the definition of thermal conductivity, suppose that a quantity of heat Q passes in the time T through an area A of a plate of conductivity k and thickness x, the sides of which are constantly maintained at temperatures B' and 8".
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.
Trans., A., 1893) that this frequency may be closely represented by the curve whose equation is y = O.21 122 5 x-( 332 (7.3 2 53 - x) 3.142.
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.
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.
Unless the initial conditions be adjusted so as to make A = o exactly, x will ultimately increase indefinitely with t.
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.
Volume =height X area of end =length of edge X area of cross-section; the " height " being the perpendicular distance between the two ends.
We have also the geometrical relations x = (a/c) (qz ry), 5 = (a/c) (rx p1), = (a/c) (pyqx).
We have F(n+i,x) -F(n,x) = (y +n)(y+n+I) F (n+2,x), whence we obtain F(i,x) _ i / y (y +I) x /(y+I)(y+2) which may also be written y +7+I-1-7+2+ - By .
We have next to determine the value of x in terms of the action between one particle and another.
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.
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.
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.
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.
We might make this investigation more general by introducing a force X as in the investigation for air, but it hardly appears necessary.
We recognize the right-hand member as the work done by the force X on the particle as the latter moves from the position x0 to the position Xi.
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 take these as axes of x and y; then if f, n be the component rotations about them, we have -
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.
We will suppose that the extraneous forces consist of a known force (X, Y, Z) at the centre, and of the reactions (Fi, Ff, R) at the point of contact.
When a liquid is in thermal and dynamical equilibrium with its vapour, then if p' and x' are the values of p and x for the vapour, and po and Xo those for the liquid, x' - xo=JL - p(I/p' - I/pc),.
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.
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.
When the correspondence of the commissioners was sent to the United States Congress the letters "X," "Y" and "Z," were inserted in place of the names of the French agents with whom the commission treated - hence the "X Y Z Correspondence," famous in American history.
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?
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'.
When the liquid is in contact with a rare medium, such as its own vapour or any other gas, x is greater than xo, and the surface energy is positive.
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.
X 67rapSos) was given by the ancients to an animal supposed to have been a cross between a lion (Lat.
X Cygni is another star of this class, remarkable for its range of magnitude.
X is the mirror rotating about the point E, and placed so that (if EB is the horizontal direction in which the rays are to be reflected) (I) the normal CE to the mirror is jointed to BC at C and is equal in length to BE, (2) the rod DBC passes through a slot in a rod ED fixed to, and in the plane of, the mirror.
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.
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.