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z

z

z Sentence Examples

  • To unlock Indiana Jones and the Fate of Atlantis, hold the Z button, then press A, the UP direction twice, then B, the DOWN direction twice, LEFT, RIGHT, LEFT direction, then B at the main menu.

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  • In that case, always try to have at least one rise or dip available for those pieces and don't worry because once you place a 'Z' piece another rise or dip becomes available to use if you need.

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  • While it's probably not advisable to pick a career based solely on where it falls in the alphabet, it can be helpful to see a full list of careers from A to Z when you're considering the available options.

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  • Often, teen fashion is influenced by what the popular figures of the day are, so when Jay Z shows up in thick, black glasses, you can expect that many teens will run to their parents, clamoring for the look!

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  • Natural diamond color varies and the highest quality white diamonds should appear colorless or clear.The GIA color grading scale rates diamond color from D (no color) thru Z (rich colors / fancy colored diamonds).

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  • The critical buzz, as well as the fan interest, around My Morning Jacket had been building slowing over the years, and in 2005, they released the album that really put them on the map in the mainstream - Z.

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  • Personal and fun, these rings can be a luxurious or inexpensive gift, depending upon the details of the piece - and initial rings can be completely personalized as they come in all 24 letters of the alphabet; A to Z.

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  • z.

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  • Curtius, Studien Z.

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  • The amount of this pressure is regulated by the screw z'.

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  • (Photos from Plates Viii., Ix., and X., P. Z.

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  • Z.

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  • On the sources of Theodoret's church history see Jeep, Quellenuntersuchungen z.

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  • } z ? ??

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  • Durand de Maillane, Dictionnaire du droit canonique (1761); Dictionnaire ecclesiastique et canonique, par une societ y de religieux (Paris, 1765); Z.

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  • Hydroplzytes and hemi-hydrophytes (aquatic plants).Of marine hydrophytes, there are, in this country, only the grass-wracks (Zostera marina and Z.

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  • spiralis, Zannichellia maritima, Z.

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  • in length and z in.

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  • Griinbaum, Neue Beitrcige z.

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  • Two of Leroy-Beaulieu's works have been translated into English: one as the Empire of the Tsars and the Russians, by Z.

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  • 757, after driving out Beornred, who had succeeded a few months earlier on the murder of A z Ethelbald.

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  • Finke, Konzilienstudien z.

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  • in Studien z.

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  • All civil cases involving less than z oo roubles value were within their competence, and more important cases by consent of the parties.

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  • d 2 for the area a, 4 27-T = 4 plae = plird2e, so that T = z pd2le.

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  • With a more direct course, and in a widening, fertile valley it continues past Downton, Fordingbridge and Ringwood, skirting the New Forest on the west, to Christchurch, where it receives the Stour from the west, and 2 z m.

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  • AS, the Roman unit of weight and measure, divided into 12 unciae (whence both "ounce" and "inch"); its fractions being deunx i 2, dextans, dodrans 4 i bes 3, septunx T7-2-, semis z, quincunx A, triens 3 i quadrans 4, sextans s, sescuncia $, uncia r i g.

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  • Its north-eastern extremity, Cape Sidero, is distant about 1 z o m.

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  • 'hlatherv i W A WoodVillef w Scale, 1:2,200,000 English Miles 20 30 40 Longitude Nest gi of Greenwich z fishery on the reefs in the Sound, much developed since 1880.

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  • 1 It has also been pointed out that the employment of the sign PI for wa and the use of z for s, cited in support of the earlier date, survived in the Kassite period.

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  • z The terms of the dedication of this book to a certain Vigilius make it impossible that the pope (538-555) of that name is meant.

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  • z, Penis.

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  • z, Caudal appendage.

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  • z, Ovo-testis.

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  • The blastopore now closes along the middle part of its course, which coincides z s FIG.

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  • This art and those of frescoand vase-painting and of gem-engraving stood higher about the z 5th century B.C. than at any subsequent period before the 6th century.

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  • At Z is the treasury of St Mark, which was originally one of the towers belonging to the old ducal palace; E, site of old houses; G, clocktower; H, old palace of procurators; J, old library; M, two columns; N, Ponte della Paglia; 0, Bridge of Sighs; W, Giants' Staircase; X, sacristy of St Mark; Y, Piazzetta.

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  • z Athan.

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  • 4 of 1846; Jamaica, z Vict.

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  • k n (S, -chunb a° Kwang`Chen ?,?z - `; r' 4; .

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  • Geschichte von Basel (3 vols., 1869-1882); Festschrift z.

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  • It owes its importance to the iron mines in the mountain Malmberget 4 z m.

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  • CC13+C02 O?OIi O / O / (4) Cl2HC CO CHCl2+CH302C CCl2C02CH3 (5) Cl2HC CONH2 Cl (z) (2) When phenol is oxidized in acid solution by chlorine, tetrachlorquinone is obtained, a compound also obtainable from hydroquinone.

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  • Gunning (Z.

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  • Dafert (Z.

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  • If triple bonds, q in number, occur also, and the energy of such a bond be Z, the equation for H becomes H = nE-+-mn -1-p(2X - Y) +q(3X - Z).

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  • Thomsen deduces the actual values of X, Y, Z to be 14.71, 13.27 and zero; the last value he considers to be in agreement with the labile equilibrium of acetylenic compounds.

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  • i.1 had read in Marcion's copies, as it does in most ancient authorities, "To the saints which are at Ephesus"; but in fact the words Ev E46w of verse z were probably absent.

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  • Further points of difficulty in connexion with the sibilants are discussed under X and Z.

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  • z) readily passes° into r in many languages: compare the Eng.

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  • The voiced sound to this is generally written z as in azure, but sometimes s as in pleasure.

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  • antiken Herrscherkulte " in Beitrdge z.

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  • z = k.si Mec ste Tha PER Deli,?

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  • r Z 1 / ?/.11 ' V o 1 J deer C ?

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  • The question, however, made little progress in parliament for some years, though Buxton, William Smith, Lushington, Brougham, Mackintosh, Butterworth, and Denman, with the aid of Z.

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  • xlvii.; Z.

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  • A relation (R) is serial when (I) it implies diversity, so that, if x has the relation R to y, x is diverse from y; (2) it is transitive, so that if x has the relation R to y, and y to z, then x has the relation R to z; (3) it has the property of connexity, so that if x and y are things to which any things bear the relation R, or which bear the relation R to any things, then either x is identical with y, or x has the relation R to y, or y has the relation R to x.

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  • ssll Laris z Ikl?

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  • au Z we' Ktisen ei e ?senfels .

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  • Krumau is principally celebrated because its ancient castle was long the stronghold of the Rosenberg family, known also as pani z ruze, the lords of the rose.

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  • Brezan, Zivot Vilema z Rosenberka (Life of William of Rosenberg), 1847; also Zivot Petra Voka z Rosenberka (Life of Peter Vok of Rosenberg), 1880.

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  • The coagulum is next flattened out by a wooden or iron roller to get rid of the cavities containing watery liquid, and the sheets are then hung up for fourteen days to dry, when they weigh about 2 lb, the sheets being usually z to a in.

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  • z, Extremity of alimentary tube.

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  • (After Hancock.) M, Ventral, Z, Extremity of intestine.

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  • z, Anal aperchief centres still remain in the ectoderm, and the fibrils form e, Heart.

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  • expression for the determinant becomes Z(-) k aitia2aa3y...anv, viz.

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  • In the theory of surfaces we transform from one set of three rectangular axes to another by the substitutions 'X=' by+ cz, Y = a'x + b'y + c'z, Z =a"x+b"y-l-c"z, where X 2+Y2+Z2 = x2+ y2+z2.

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  • Further we find x=aX+a'Y+a"Z, y=bX z= cX+c'Y+ c"Z, and the problem is to express the nine coefficients in terms of three independent quantities.

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  • � Oxl d 2x 77n If we have new variables z such that zs=4s(yl, Y2,...yn), we have also z s =1 Y 8(x1, x2,���xn), and we may consider the three determinants which i s 7xk, the partial differential coefficient of z i, with regard to k .

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  • yl, y 2,...yn) (zl, z2,...zn z1, z 2, ���zn xi, 'X' 2,...

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  • =0, a'xn+ (b'y+c'z)xn-1+...

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  • = 0, we find that, eliminating x, the resultant is a homogeneous function of y and z of degree mn; equating this to zero and solving for the ratio of y to z we obtain mn solutions; if values of y and z, given by any solution, be substituted in each of the two equations, they will possess a common factor which gives a value of x which, corn bined with the chosen values of y and z, yields a system of values which satisfies both equations.

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  • If three equations, each of the second degree, in three variables be given, we have merely to eliminate the six products x, 2, z 2, yz, zx, xy from the six equations u = v = w = o = oy = = 0; if we apply the same process :to thesedz equations each of degree three, we obtain similarly a determinant of order 21, but thereafter the process fails.

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  • X (1 +PD1+12D2+...+�8D8+...) fm, and now expanding and equating coefficients of like powers of /t D 1 f - Z(Difi)f2f3.

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  • = a l z = e a2z =e.3.=...

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  • (1 + a i x) (1+ = s i z we have the symbolic identity +02712+0.3x3+...

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

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  • Moreover, instead of having one pair of variables x i, x2 we may have several pairs yl, y2; z i, z2;...

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  • We frequently meet with cogredient and contragedient quantities, and we have in general the following definitions:-(i) " If two equally numerous sets of quantities x, y, z,...

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  • x', y', z',...

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  • are such that whenever one set x, y, z,...

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  • is expressed in terms of new quantities X, Y, Z, ...

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  • the second set x', y', z', ...

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  • is expressed in terms of other new quantities X', Y', Z', ....

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  • (2) " Two sets of quantities x, y, z, ...; E, n, i, ...

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  • are said to be contragredient when the linear substitutions for the first set are x =A1X+u1Y-}-v1Z-?--..., y = A2X+,u2Y +v2Z �..., Z = A 3 X +�3Y -1v 3 Z - -..., and these are associated with the following formulae appertaining to the second set, .`"?.

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  • + ���, Z = v16+v2%/+v3" +���, wherein it should be noticed that new quantities are expressed in terms of the old, as regards the latter set, and not vice versa."

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  • The symbols - dy, d z, ...

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  • are contragredient with the d- variables x, y, z, ...

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  • for when (x, z, ���) = (A l, �i, VI I ���) (X, Y, Z, ���), I A 2, / 2 2, Y2, ...

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  • rd Y' ' ...) = 01, A2, A 3, ...) (d ' ' z / 2 1, /22, / 1 3, ...

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  • If u, a quantic in x, y, z, ..., be expressed in terms of new variables X, Y, Z ...; and if, n,, ..., be quantities contragredient to x, y, z, ...; there are found to exist functions of, n, ?, ..., and of the coefficients in u, which need, at most, be multiplied by powers of the modulus to be made equal to the same functions of E, H, Z, ...

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  • -2 _ ab 2an-2bn-2Crz z x () x x x, Each term on the right-hand side may be shown by permutation of a, b, c to be the symbolical representation of the same covariant; they are equivalent symbolic products, and we may accordingly write 2(ac) (bc)ai -1 bi -1 cx 2 =(ab)2a:-2b:-2c:, a relation which shows that the form on the left is the product of the two covariants n (ab) ay 2 by 2 and cZ.

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

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

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  • In fact, if f and 4, have these forms, it is easy to verify that (f, 4,)i= (A j z) (rs)r x s x .

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  • lii.) that the substitution, z =?

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  • When a z and the invariants B and C all vanish, either A or j must vanish; in the former case j is a perfect cube, its Hessian vanishing, and further f contains j as a factor; in the latter case, if p x, ax be the linear factors of i, f can be expressed as (pa) 5 f =cip2+c2ay; if both A and j vanish i also vanishes identically, and so also does f.

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  • l aa k -x 2 d d- = 0; Z(nk)ak+l adk - x ldd2=0; or in the form d d 52-x 2(7 =0, O - x1ax2 = 0; where 0 = ao d a l + 2a 1 -?...+na,,_id an, 0 = nal dao -?

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

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

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  • - zn +9 1 -z2.1 -z3....1-z8; and since this expression is unaltered by the interchange of n and B we prove Hermite's Law of Reciprocity, which states that the asyzygetic forms of degree 0 for the /t ie are equinumerous with those of degree n for the The degree of the covariant in the variables is e=nO-2w; consequently we are only concerned with positive terms in the developments and (w, 0, n) - (w - r; 0, n) will be negative unless nO It is convenient to enumerate the seminvariants of degree 0 and order e=n0-2w by a generating function; so, in the first written generating function for seminvariants, write z2 for z and az n for a;.

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  • we obtain 1 - z - 2 1 -az".

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  • As we have to do only with that part of the expansion which involves positive powers of z, we must try to isolate that portion, say A n (z).

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  • the complete function may be written ll A2(z) i 2A2 (z/ ' A 2 z 1az2 1.1-a2; and this is the reduced generating function which tells us, by its.

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  • Again, for the cubic, we can find A3(z) - -a6z6 1 -az 3.1 -a 2 z 2.1 -a 3 z 3.1 -a4 where the ground forms are indicated by the denominator factors, viz.: these are the cubic itself of degree order I, 3; the Hessian of degree order 2, 2; the cubi-covariant G of degree order 3, 3, and the quartic invariant of degree order 4, o.

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  • Similarly for the quartic A 4 /z) - -a s z 1 -az4.1 -a2.1-a2z4.1-a3 .1 establishing the 5 ground forms and the syzygy which connects them.

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  • For these two cases the perpetuants are enumerated by z 2 23 -z2' and l -z2.1-z3 respectively.

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  • These latter forms are enumer ated by I - z 24 I -z 4; hence the generator of quartic perpetuants must be z4 z4 z7 1-z 2.1 -z 3.1z 4 1-z 2.1-s 4 1-22.1-z3.1-z4' and the general form of perpetuants is (4 K+ 1 3A+1 2�).

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  • The Number Of Linearly Independent Seminvariants Of The Given Type Will Then Be Denoted By (W; 0, P; 0', Q) (W; 0, P; 0', Q); And Will Be Given By The Coefficient Of A E B E 'Z W In L Z 1 A.

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  • 1 Bz4' That Is, By The Coefficient Of Z W In Zp '.

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  • 1 Zp 0.1 Z 4 1.1 Z 2.

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  • 1 Z4 E, 1 Z.

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  • 1 Z 2.1 Z 3 ....

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  • 1 Z 0.1 Z 2.1 Z 3 ....

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  • Taking The First Generating Function, And Writing Az P, Bz4, 2 For A, B And Z Respectively, We Obtain The Coefficient Of Aobe'Zpo 0' 2W That Is Of A E B E 'Z �, In 1 Z 2 1 Azp. 1 Azp 2....1 A2 P 2.1 Az P .

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  • Thus, For Two Linear Forms, P =Q = I, We Find 1 Z 2 1 Az.

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  • For The Degrees I, 2, The Asyzygetic Forms Are Enumerated By Z.

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  • 1 And The Actual Forms For The First Three Weights Are 1 Aobzo, (Ao B 1 A 1 B O) Bo, (A O B 2 A 1 2 0 Bo, Ao(B2, 3 A1B2 A2B1 A O (B L B 2 3B O B 3) A I (B 2 1 2B 0 B 2); Amongst These Forms Are Included All The Asyzygetic Forms Of Degrees 1, 1, Multiplied By Bo, And Also All The Perpetuants Of The Second Binary Form Multiplied By Ao; Hence We Have To Subtract From The 2 Generating Function 1Z And 1 Z Z2, And Obtain The Generating Function Of Perpetuants Of Degrees I, 2.

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  • 1 _ 1 _ Z 2 Z3 1 Z.

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  • 1 Z 2 1 Z 1 Z 2 I Z.

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  • 1 Z 2 ....

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

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  • For the case 8=1, 0' =2, the condition is a i r 1 72 = A032=0; and the simplest perpetuant, derived directly from the product A 1 B 21 is (I)a(2)b-(21)b; the remainder of those enumerated by z3 I z.

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  • 1 - z 2.1 -z 3.1 - z4' The series may be continued, but the calculations soon become very laborious.

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

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  • They consist of a square citadel (Bairam Ali Khan kalah), s z m.

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  • If V denote the potential, F the resultant force, X, Y, Z, its components parallel to the co-ordinate axes and n the line along which the force is directed, then - sn = F, b?= X, - Sy = Y, -s Surfaces for which the potential is constant are called equipotential surfaces.

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  • of magnetization is z = S aK approximately, E l being the electrochemical equivalent and S the density of the metal.

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  • In a gorge 1 z m.

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  • 11 Z.

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  • The papyrus, which is of the 3rd century, was discovered by Bickell among the Rainer collection, who characterized it (Z.

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  • On the language of Joel, see Holzinger, Z.

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  • Thurnlach, Annalen, 1882, 213, p. 369) th: - Z us n(C H) H20 CH �CHO - > CH3�CHCH3�CH

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  • z The A.S.

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  • z Cranmer's works are to be found in Burnet, " Collection of Records " appended to his History of the Reformation (ed.

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  • Obrucheff, Central Asia, Northern China and the Nanshan (1900-1901); Z.

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  • z Subsequently extended till 1907.

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  • Nevy, Z.

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  • (b) If n = a z, then x =tog a n.

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  • the number of products such as x r, xr-3y3, x r-2 z 2,.

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  • that can be formed with positive integral indices out of n letters x, y, z,..

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  • z considerable use of anatomical characters in his definitions of larger groups, and may thus be considered as the father of modern zoology.

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  • Taking co-ordinates in the plane of the screen with the centre of the wave as origin, let us represent M by, n, and P (where dS is situated) by x, y, z.

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  • This integral is the Bessel's function of order unity, defined by J,(z) n (z cos 0) sin 24 d4)..

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  • Thus, if x = R cos 4), C =,r2R2J1(pR) pR and the illumination at distance r from the focal point is 4T2 r 21rRr1 fX (2 fKr) a J The ascending series for J 1 (z), used by Sir G.

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  • Trans., 1834) in his original investigation of the diffraction of a circular object-glass, and readily obtained from (6), is z z 3 25 27 J1(z) = 2 2 2.4 + 2 2.4 2.6 2 2.4 2.6 2.8 + When z is great, we may employ the semi-convergent series Ji(s) = A/ (7, .- z)sin (z-17r) 1+3 8 1 ' 6 (z) 2 3.5.7.9.1.3.5 5 () 3 1 3.5.7.1 1 3 cos(z - ?r) 8 ' z (z) 3.5.7.9.11.1.3.5.7 1 5 + 8.16.24.32.40 (z

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  • Now by definition J (z) _ C cos (z cos e) do = 1 - 22-%2?

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  • 12 -7r2R4 x2 f 2 The roots of Jo(z) after the first may be found from We may compare this with the corresponding result for a rectangular aperture of width a, tlf = X/a; and it appears that in consequence of the preponderance of the central parts, the compensation in the case of the circle does not set in at so small an obliquity as when the circle is replaced by a rectangular aperture, whose side is equal to the diameter of the circle.

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  • When z has one of the values thus determined, (z)=Jo(z).

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  • The accompanying table is given by Lommel, in which the first column gives the roots of J2(z) =o, and the second and third columns the corresponding values of the functions specified.

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  • We have 12 = 71-212.4 4J12(z) X 2 f 2 z2 z = 21rRr/X f (20).

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  • Thus so that z 1 J1 2 (z) = - 2 Jo 2 (z) - qz.h2(Z), (' an n z 1 J i 2 (z)dz = 1 -Jo (z) - J 1 2 (z).

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  • If r, or z, be infinite, Jo(z), J 1 (z) vanish, and the whole illumination is expressed by 71-R 2, in accordance with the general principle.

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  • In any case the proportion of the whole illumination to be found outside the circle of radius r is given by J02(z)+J12(z).

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  • For the dark rings Ji(z) =o; so that the fraction of illumination outside any dark ring is simply Jo 2 (z).

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  • When z is great, the descending series (io) gives i 2J 1 (z) = 2 sin (z1 7r) 22; (2) z so that the places of maxima and minima occur at equal intervals.

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  • It may be instructive to contrast this with the case of an infinitely narrow annular aperture, where the brightness is proportional to Jo 2 (z).

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  • When z is great, J°(z) = ()cos (z '-hir).

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  • The mean brightness varies as .51; and the integral f J02(z)zdz is not convergent.

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  • Ein mikroskopischer Gegenstand z.

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  • J 1 2 (z)dz.

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  • Now by (17), (18) z 1 J1(z) =Jo(z)- (z); where P increases.

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  • The extreme value possible for a is a right angle, so that for the microscopic limit we have Z X o/µ (2).

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  • If x and y be co-ordinates in the plane of the wave-surface, the axis of y being parallel to the lines of the grating, and the origin corresponding to the centre of the beam, we may take as an approximate equation to the wave-surface -- -} z =+Bxy 2, +ax 3 13x2 2pp p y+-yxy2-?-Sy3+..

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  • The constant multiplier is of no especial interest so that we may take as applicable to the image of a line 0 I = z 2 sin e A f 1+cos ` - 271 - Eh).

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  • In the second term if we observe that cos {p'+ 27rh/Af)E} =cos{p' - g,E} = cos p cos g, +sin p sin giE, we see that the second part vanishes when integrated, and that the remaining integral is of the form w = f +.0 sin z h, cos where h,=7rh/Af, g,=a-27Th/Af.

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  • The formation of bands thus requires that the retarding plate be held upon the side already specified, so that zs be positive; and that the thickness of the plate (to which z is proportional) do not exceed a certain limit, which we may call 2T 0.

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  • 2 l v i.,,rv2 C+iS= o e .irfo e dv3; and, by continuing this process, - -iS Z 1"J2 v3% 2 3 J?v S 2 2 v 7 -}- .

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  • Comparing the expressions for C, S in terms of M, N, and in terms of G, H, we find that G = z (cos u+sin u)-M, H = z (cos u-sin u) +N.

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  • (19), 1 abA) ' ' we may write 12= (cos 27rv 2 .dv) 2 + (f sin zirv 2 .dv) 2 (20), or, according to our previous notation, 12 = (2 - C 2 +(z - Sv)2= G2 +H2 Now in the integrals represented by G and H every element diminishes as V increases from zero.

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  • The co-ordinates of J, J' being (- z, - z), I 2 is 2; and the phase is, period in arrear of that of the element at 0.

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  • 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.

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  • 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).

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  • If we suppose that the force impressed upon the element of mass D dx dy dz is DZ dx dy dz, being everywhere parallel to the axis of Z, the only change required in our equations (I), (2) is the addition of the term Z to the second member of the third equation (2).

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  • (b2V2 + n2) (a2 - b 2) = - z It will now be convenient to introduce the quantities a l, a 2', 7731 which express the rotations of the elements of the medium round axes parallel to those of co-ordinates, in accordance with the equations Ty - 1 = dz ' 3= - dy 2 = dx - In terms of these we obtain from (7), by differentiation and subtraction, (b 2 v 2 + n 2) 7,3 = 0 (b 2 0 2 +n 2) .r i = dZ/dy (b 2 v 2 +n 2)', , 2 = - dZ/dx The first of equations (9) gives 3 = 0 (10) For al we have ?1= 47rb2, f dy e Y tkr dx dy dz

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  • Integrating by parts in (II), we get J e = ikr d7 pc-11 / d (e r - ay= rJ Z d y - r / 1 dY, in which the integrated terms at the limits vanish, Z being finite only within the region T.

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  • Thus f (= 4-rb 2;JJ Z dY (e r) dx dy dz.

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  • 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 ?

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  • - 47rb 2 dr (r / ' denoting the angle between r and z.

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  • This displacement, which we may denote by; is in the plane containing z and r, and perpendicular to the latter.

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  • By all real estate deeds the sale of intoxicating liquors is for ever prohibited in the city; and an act of the state legislature in 1909 prohibited the sale of intoxicating liquor within r z m.

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  • After leaving the Central Provinces, the river widens out in the fertile district of Broach, with an average breadth of z m.

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  • It lies partly within Stillwater and partly within Half-Moon townships, in the bottom-lands at the mouth of the Anthony Kill, about 1 z m.

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  • 2 The date of L and Z is given as the end of the 15th century in the introduction to Wyse's edition.

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  • Delitzsch., Beitrdge z.

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  • Teichmuller, Studien z.

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  • 197; Jiirgeltinas, " Protective Action of Granulations," Beitrcige z.

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  • Z 0 Huntonbridge oSarratt .,Micklefieltl Green 4.E ' Gt(2p,steado .T9961111 ?

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  • rocks the working-places may sometimes exceed z oo or even 200 ft.

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  • Z FIG.

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  • Z?:C -?

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  • 832-834, and a good exposition may be found in Z.

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  • in diameter, and vary in thickness from a to z in., the centre being the thickest.

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

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  • As an example of the general equations, take the simplest case of a uniform field of gravity, with Oz directed vertically downward; employing the gravitation unit of force, 1 dp i dp t dp dp/dz = p = pzn (4) z n+I pz 1 /n p-p n-H ?t), (5) supposing p and p to vanish together.

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  • From the gas-equation in general, in the atmosphere n d dp _ I dp 1 de _ d0 de i de (8) z p dz-edz-p-edz-k-edz' which is positive, and the density p diminishes with the ascent, provided the temperature-gradient de/dz does not exceed elk.

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  • With uniform temperature, taking h constant in the gas-equation, dp / dz= =p / k, p=poet/ k, (9) so that in ascending in the atmosphere of thermal equilibrium the pressure and density diminish at compound discount, and for pressures p 1 and 1, 2 at heights z 1 and z2 (z1-z2)11?

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  • The time rate of increase of momentum of the fluid inside S is )dxdydz; (5) and (5) is the sum of (I), (2), (3), (4), so that /if (dpu+dpu2+dpuv +dpuw_ +d p j d xdyd z = o, (b)` dt dx dy dz dx / leading to the differential equation of motion dpu dpu 2 dpuv dpuv _ X_ (7) dt + dx + dy + dz with two similar equations.

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  • dp dpu dpv dpw -z)' reducing to the first line, the second line vanishing in consequence of the equation of continuity; and so the equation of motion may be written in the more usual form du du du du d dt +udx+vdy +wdz =X -n dx' with the two others dv dv dv dv i dp dt +u dx +v dy +w dz - Y -P d y' dw dw dw Z w dw i d p dt +u dx +v dy +wd - -P dz.

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  • 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.

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  • If F (x, y, z, t) =o represents the equation of a always the same particles of fluid, DF =o, or dF {-u dx-rzd { w d d = o, Trt y _ which is called the differential equation of the bounding surface.

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  • But in the most general case it is possible to have three functions 0, ?, m of x, y, z, such that udx+vdy +wdz = -dcp-mdl,G, (I) as A.

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  • Clebsch has shown, from purely analytical considerations (Crelle, lvi.); and then = Z d(?G, m), ?

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  • = 1d(1G, m),?= 1 d(?', m) d(y, z) d(z, x)' y), and, a ?+n d +s m - O,, d +n dinn +?

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  • If homogeneous liquid is drawn off from a vessel so large that the motion at the free surface at a distance may be neglected, then Bernoulli's equation may be written H = PIP--z - F4 2 / 2g = P/ p +h, (8) where P denotes the atmospheric pressure and h the height of the free surface, a fundamental equation in hydraulics; a return has been made here to the gravitation unit of hydrostatics, and Oz is taken vertically upward.

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  • In particular, for a jet issuing into the atmosphere, where p=P, q 2 /2g = h - z, (9) or the velocity of the jet is due to the head k-z of the still free surface above the orifice; this is Torricelli's theorem (1643), the foundation of the science of hydrodynamics.

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  • For instance, in a uniplanar flow, radially inward towards 0, the flow across any circle of radius r being the same and denoted by 27rm, the velocity must be mfr, and 0=m log r,, y=m0, +4,i =m log re ie, w=m log z.

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  • 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?

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

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  • Example 3.-Analysing in this way the rotation of a rectangle filled with liquid into the two components of shear, the stream function 1//1 is to be made to satisfy the conditions v 2 /1 =0, 111+IRx 2 = IRa 2, or /11 =o when x= = a, +b1+IRx 2 = I Ra2, y ' 1 = IR(a 2 -x 2), when y = b Expanded in a Fourier series, 2 232 2 cos(2n+ I)Z?rx/a a -x 7r3 a Lim (2n+I) 3 ' (1) so that '?"

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  • ch (2n +1)I 7 ry /a yl-R3ct (2n +I)3.ch(2n+I)17b /a ' 16 cos(2n+I) 2 7 z /a w1=4,i+ 4, ii = iR ?3a2 an elliptic-function Fourier series; with a similar expression for 1,'2 with x and y, a and b interchanged; and thence 4, = '1 +h.

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  • z) = I (ax - I d ?

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  • The velocity past the surface of the sphere is dC r sin 0 dy 2U (2r+ a 2) r sin g z U sin e, when r =a; (20) so that the loss of head is (!

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  • With a velocity function 49, the flow -f d 4 = 4)142, (2) (9) (to) (6) (22) Z Uy (I -a4,ic /r4), so that the flow is independent of the curve for all curves mutually reconcilable; and the circulation round a closed curve is zero, if the curve can be reduced to a point without leaving a region for which 4 is single valued.

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  • Employing the equation of continuity when the liquid is homogeneous, 2 (cly - d z)?

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  • 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.

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  • dt-(u)dy- (w-w) dz = d - (U-yR+zQ) dy - (V-zP+xR)d -(W-xQ+yP) d z (8) is the time-rate of change of 49 at a point fixed in space, which is left behind with velocity components u-u', v-v', w-w'.

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  • Z, v' -V --zP - x R b2 2b2 c2 S2 1 z a2+ 2b2S23x, w' = w -{-xQ-yP2c2 cl?

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  • /c - a2, 1 z a = (c2 +a2)2 "2 +a21 J z z 2 ?

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  • z 2 a - 1 / 4 (a 2 +b 2) 2) ?

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  • In the general motion again of the liquid filling a case, when a = b, 52 3 may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce to d =- 2+ 2 J, = 2 x22111, d = 2 2`2 (+/'15-Om) (1 yy y n`t dt a +c dt a +c dt a +c) dc2, a2-1-c2 d122 a2 c2 dt ="2) +a2= G2y 71' dt = 121 1 - a 2 -c 2SJ, (19) of which three integrals are e +777 r z y 2= L -?2J2, (20) (a2 + c2) 2 2 121+14 =M+ 2c2(a2-c2)1 ' (21) 121+522hN = + x24 2,2 and then (dt / 2 = (a2 + c 2) 2(° v 2 - 12171) 2 4C4 2 2 - (+ c2)2?(E+77) (?

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  • Put S2 1 =12 cos 4, 12 2 = -12 sin 4, d4 d52 1 dS22 Y a2+c2 122 7Ti = 71 22 CL2- c2(121+5221)J, a2 +c2 do a2+c2 + 4c2 z dt a'-c2 (a2+,c2)2 M+2c2(a2-c2 N-{-a2+c2 2 Ý_a 2 +c 2 (' 4c2 .?"d za 2 -c 2 c2)2 2'J Z M+ -c2) which, as Z is a quadratic function of i 2, are non-elliptic so also for; G, where =co cos, G, 7 7 = - sin 4.

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

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  • This is a closed Sumner line for n =I, when the boundary consists of two parallel walls; and n= z gives an Elastica.

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  • Beginning with a single body in liquid extending to infinity, and denoting by U, V, W, P, Q, R the components of linear and angular velocity with respect to axes fixed in the body, the velocity function takes the form = Ucb1+V42+W43+ P xi+Qx2+Rx3, (I) where the 0's and x's are functions of x, y, z depending on the shape of the body; interpreted dynamically, C -p0 represents the impulsive pressure required to stop the motion, or C +p4) to start it again from rest.

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  • 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.

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

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  • , (4) where X, Y, Z, L, M, N denote components of external applied force on the body.

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  • These equations are proved by taking a line fixed in space, whose direction cosines are 1, then dt=mR-nQ,' d'-t = nP =lQ-mP. (5) If P denotes the resultant linear impulse or momentum in this direction P =lxl+mx2+nx3, ' dP dt xl+, d y t x2' x3 +1 dtl dt 2 +n dt3, =1 ('+m (dt2-x3P+x1R) ' +n ('-x1Q-{-x2P) ' '= IX +mY+nZ, / (7) for all values of 1, Next, taking a fixed origin and axes parallel to Ox, Oy, Oz through 0, and denoting by x, y, z the coordinates of 0, and by G the component angular momentum about 1"2 in the direction (1, G =1(yi-x2z+x3y) m 2-+xlz) n(y(y 3x 1 x3x y + x 2 x) (8) Differentiating with respect to t, and afterwards moving the fixed.

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  • Kirchhoff's expressions for X, Y, Z, the coordinates of the centre of the body, FX=y 1 cos xY--y 2 cos yY-{-y 3 cos zY, (18) FY = -y l cos xX -Hy2 cos yX+y 3 cos zX, (Ig) G=y 1 cos xZ+y 2 cos yZ+y 3 cos zZ, (20) (21) F(X+Yi) = Fy3-Gx3+i /) X 3epi.

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  • (6) The least admissible value of p is that which makes the roots equal of this quadratic in µ, and then ICI s ec 0,, u= z - p (7) the roots would be imaginary for a value of p smaller than given by Cip 2 - 4(c 2 -c i)c2C 2 u 2 =o, (8) p2 = 4(c 2 -c l)cl C2.

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  • Y I /t= (a - (3)M'QR cos o+ER= (a - (3)M'UR+0R, (7) Z = lt.

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  • sin o= F dl, (20) C3 do F2 h _ F2 cos 2 o F 2 sin z o F dt y - V C G c +2 c1 coso+H]; (21) 1 z so that cos 0 and y is an elliptic function of the time.

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  • ait Z?

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  • Land which contains less than about z /c, of lime usually needs the addition of this material.

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  • is the normal for ordinary manufacturing and smoking tobaccos, 1 to i z ft.

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  • 2 lb of zinc in the form of sulphate, and z to 4 oz.

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  • MIDSOMER NORTON, an urban district in the northern parliamentary division of Somersetshire, England, i 2 z m.

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  • '0,1tfill', Z -4,'If';' ' sel..Vk Q7Vra ----4‘kiar,, 'e/-ke;,/ ¦.

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  • z Compare the interesting résumé of the whole question in Boyd Dawkins's Early Man in Britain (London, 1880).

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  • Deschales (Cursus, seu Mundus Mathematicus, 1674); Z.

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  • bank of the Liris with a suburb on the opposite bank 1 z m.

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  • p, Lips of redia; q, collar; r, processes serving as rudimentary feet; s, embryos; 1, trabecula crossing body-cavity of redia; u, glandular cells; v, birth-opening; w, w', morulae; y, oral sucker; y', ventral sucker; z, pharynx.

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  • Forests cover nearly r z million acres, yielding valuable timber (teak, sandalwood, &c.), and affording grazing-ground for cattle.

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  • tachinoides, to about i i z millimetres in that of G.

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  • In both cases, it need hardly be said, the great literary and spiritual value of the later passages ought in no way 1 Regarded by Stade (Z.

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  • m., and its population (1905) to 2,009,320, of whom about 60% are Roman Catholics, 37% Protestants, 1 z% Jews, and the remainder of other confessions.

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  • z This account is at variance with the literary evidence and rests on that of the coins, as set forth by I.

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  • mucosus the Indian "rat-snake," Z.

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