# P sentence example

He'd better watch his

**P**'s and Q's.Every industrial concern employing fifty hands or over elects one or more delegates to the electoral

**P**?Among the Terebelloidea there is a remarkable differentiation of the ne

**p**hridia into two series.Let AB be the major axis of the orbit, B the pericentre, F the focus or centre of motion,

**P**the position of the body.One of the most

**p**im ortant ways of keeping insect pests in check is by " spraying " or " washing."AdvertisementBut in his character as phenomena must be examined or what may be neglected

**p**y g in economic inquiry.**P**, Pedal ganglion with otocyst attached.In the district, and es

**p**ecially at Salobrena, 3 m.The same deficiency became still more apparent when, between 1869 and 1871, he published his Hand-List of Genera and Species of Birds in three 1813-1814,

**p**. xxviii.); but, through the derangements of that stormy period, the order was never carried out (Mem.This thereat German comparative anato- Johannes great

**p**mist did in two communications to the Academy of Sciences of Berlin, one on the 26th June 1845 and the other on the 14th May 1846, which, having been first briefly published in the Academy's Monatsbericht, were afterwards printed in full, and illustrated by numerous figures, in its Abhandlungen, though in this latter and complete form they did not appear in public until 1847.AdvertisementNo doubt they all agreed in saying that they were prosecuting Y g Y g Y

**P**g a search for what they called the true system of nature; but that was nearly the end of their agreement, for in what that true system consisted the opinions of scarcely any two would coincide, unless to own that it was some shadowy idea beyond the present power of mortals to reach or even comprehend.The pedal equation is r 3 =a 2

**p**, which shows FIG.According to the system of phonetic changes generally known as "Grimm's law," an original b appears in English as

**p**, an original bh as b.The work of the committee is by no means simple, as frequently very few transactions take place

**p**?This question can readily be answered as regards the past forty years or so, for which material g

**p**y y ?AdvertisementA change of the Hebrew text seems necessary; possibly we should read S1p $t"', "low is the voice," instead of 51p$ o'

**p**', "he rises up at the voice."Its consonants are k, g, ng, ch, j, n, t, d, n,

**p**, b, m, y, r, l, w, s, h.Let

**P**,**P**' be two consecutive positions of the radius vector.It follows that

**p**must be greatest at pericentre, where its distance from F is least.By geometrical consideration it can be shown that the angle subtended by

**p**, as seen from F, must be inversely as the square of its distance r.AdvertisementRepresenting by

**P**this position, it follows that the area of that portion of the ellipse contained between the radii vectores FB and FP will bear the same ratio to the whole area of the ellipse that t does to T, the time of revolution.Di-derivatives x x x

**p**v as $ v as s Here we have assumed the substituent groups to be alike; when they are unlike, a greater number of isomers is possible.Eliminating a and b between these relations, we derive

**P**k V k /Tk= 8R, a relation which should hold between the critical constants of any substance.If, however, an amount of energy a is taken up in separating atoms, the ratio is expressible as C

**p**/C„= (5+a)/(3-Fa), which is obviously smaller than 5/3, and decreases with increasing values of a.These relations may be readily tested, for the ratio C

**p**/C„ is capable of easy experimental determination.AdvertisementMitscherlich, in the case of the acid phosphate and acid arsenate of potassium, KH 2

**P**(As)04, who adopted the term isomorphism, and regarded phosphorus and arsenic as isomorphously related elements.In Greek, where I is the twentieth letter of the alphabet, or, if the merely numerical and

**p**are excluded, the eighteenth, another form 1 or S according to the direction of the writing is also widespread.Lumby (Cambridge, 1883), supplemented a little„by Edward Hall (Chronicle,

**p****p**. 3 6 3-3 6 4).A distinguished philosopher or man of letters would find them bidding f o r his presence, and most of the great names are

**p**?Ducarla, published his La France consideree da

**p**s les dijferentes hauteurs de ses plaines (1791), upon which equidistant contours at intervals of 16 toises found a place.Of SchOner we know that he produced four globes, three printed from segments (1515, 1523, 1 533), and

**p**SCF12MER.S FIG.This map was repeatedly revised, Antgria- g

**P****P**Y ?Although many " General " and other meetings were held in different Period of parts of the country for the purpose of setting

**P**Y**P****P**g forth Quakerism, the notion that the whole Christian church would be absorbed in it, and that the Quakers were, in fact, the church, gave place to the conception that they were " a peculiar people " to whom, more than to others, had been given an understanding of the will of God.This identification can only be

**p**urchased at the cost of a complete renunciation of the Avestan genealogy.The dentition is i, c 1,

**p**4, m L total 42.Fully realizing, the difficult

**P**Y g?**P**For sulphonic acids containing nitrogen see Ammonia.When he had ruthlessly quelled the resistance offered to his accession by his brothers, who both fell in the struggle for the throne, Selim undertook his campaign in Persia, having first extirpated the Shia heresy, prevalent 5 e 12 m, g

**P**Y,**P**1512152.0.**P**Europe, after the suppression of a revolt of the governor of Damascus, who had thought to take advantage of the new sultan's accession to restore the independent rule of the Circassian chiefs.He imroved the laws and institutions established by

**p**i his predecessors and adapted them to the require ments of the age; to him are due important modifications in the feudal system, aimed.Their central organization was in Paris and their objects were known throughout Europe, but except at Yildiz Kiosk their power was

**P****P****p**almost everywhere underrated.In 734 their king Sanip(b)u was a vassal of Tiglathpileser IV., and his successor,

**P**(b)udu-ilu, held the same position under Sennacherib and Esarhaddon.Chemie, 1889,

**p**. 96; and 1892, 9,**p**.The result of the investigation shows that the electrical work Ee is given by the_equation Ee =1 where v is the volume of the solution used and

**p**its osmotic pressure.On the analogy between this case and that of the interface between two solutions, Nernst has arrived at similar logarithmic expressions for the difference of potential, which becomes proportional to log (

**P**1 /**P**2) where P2 is taken to mean the osmotic pressure of the cations in the solution, and**P**i the osmotic pressure of the cations in the substance of the metal itself.Yet later the co

**p**e seems to have been authoritatively proscribed with the rest.Tubes are generally made up around mandrels, and allowed throughout the curing to remain imbedded i n

**p**u lverized French chalk, which affords a useful support for many articles that tend to lose their shape during the process.On the opposite side of Betanzos Bay (the

**p**yas Acµl i v or Portus Magnus of the ancients) is the great port of Corunna or Coruna.Resultant Expressible as a Determinant.-From the theory of linear equations it can be gathered that the condition that

**p**linear equations in**p**variables (homogeneous and independent) may be simultaneously satisfied is expressible as a determinant, viz.Denote by brackets () and [] symmetric functions of the quantities

**p**and a respectively.**P**y q +...Writing D = gi d od

**p**!In the present particular case putting m 10 = 1 2, mot= v and m P4 =o otherwise M10t+M01n+...+Mpot

**P**n 4 +...Recalling the formulae above which connect s P4 and a m, we see that dP4 and Dp q are in co-relation with these quantities respectively, and may be said to be operations which correspond to the partitions (pq), (10

**P**01 4) respectively.We may remark the particular result (-)

**p**+**p**q!Since dp4+(-)

**P**+T1(**p**+q qi 1)!dd4, the solutions of the partial differential equation d P4 =o are the single bipart forms, omitting s P4, and we have seen that the solutions of p4 = o are those monomial functions in which the part pq is absent.Remark.-The invariant C is a numerical multiple of the resultant of the covariants i and j, and if C = o,

**p**is the common factor of i and j.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.Solving the equation by the Ordinary Theory Of Linear Partial Differential Equations, We Obtain

**P**Q 1 Independent Solutions, Of Which**P**Appertain To S2Au = 0, Q To 12 B U =0; The Remaining One Is Ab =Aobl A 1 Bo, The Leading Coefficient Of The Jacobian Of The Two Forms. This Constitutes An Algebraically Complete System, And, In Terms Of Its Members, All Seminvariants Can Be Rationally Expressed.Thus, For Two Linear Forms,

**P**=Q = I, We Find 1 Z 2 1 Az.They have continued to be worn, however, by the bishops of the Scandinavian Lutheran Y

**P**Churches.The Hebrew titles ascribe to him seventy-three psalms; the Septuagint adds some fifteen more; and later opinion, both Jewish

**p**and Christian, claimed for him the authorship of the whole Psalter (so the Talmud, Augustine and others).When Athens became the capital in 1833 the ancient name of The

**P**33 Peiraeus.If Hebrew, it might be derived from the root

**p**rr (to embrace) as an intensive term of affection.No purely astronomical enterprise was ever carried out on so Transits of

**P**large a scale or at so great an expenditure of money and labour as was devoted to the observations of these transits, and for several years before their occurrence the astronomers of every leading nation were busy in discussing methods of observation and working out the multifarious details necessary to their successful application.Since

**p**can never be infinite, complete shielding is not possible.If

**P**is the weight of the magnet, l the length of each of the two threads, 2a the distance between their upper points of attachment, and 2b that between the lower points, then, approximately, MH =**P**(ab/l) sin 0.The magnetized body which is to be tested should be placed in such a position that the force H

**P**due to its poles may, at the spot occupied by the suspended needle, act in a direction at right angles to that due to the earth - that is, east and west.The angle B is indicated by the position of the spot of light upon the scale, and the horizontal intensity of the earth's field H E is known; thus we can at once determine the value of H

**P**, from which the magnetization I of the body under test may be calculated.Denoting the distance AM by d 1, BM by d2, and AB by 1, we have for the force at M due to the magnetism of the rod H

**P**=d 12 - horizontal component (dla - d 2 3).The force acting on the magnetism of one of the faces, and urging this face towards the other, will be less than B by 27r1, the part of the total force due to the first face itself; hence the force per unit of area with which the faces would press against each other if in contact is

**P**= (B-27rI)I =27rT 2 +HI = (B 2 -H 2) =/81r.The width of the gap may be diminished until it is no greater than the distance between two neighbouring molecules, when it will cease to be distinguishable, but, assuming the molecular theory of magnetism to be true, the above statement will still hold good for the intermolecular gap. The same pressure

**P**will be exerted across any imaginary section of a magnetized rod, the stress being sustained by the intermolecular springs, whatever their physical nature may be, to which the elasticity of the metal is due.The whole of the rod will therefore be subject to a compressive longitudinal stress

**P**, the associated contraction R, expressed as a fraction of the original length, being R =**P**/M = (B 2 -H2)/87-M, where M is Young's modulus.The equites remained' at home, or only went out as members of the general's staff, their places being taken by the equites equo

**p**y ivato, the cavalry of the allies and the most skilled horsemen of the subject populations.The sperm is removed by the male from the genital aperture into a special receptacle on the terminal segment

**P**FIG.**P**.Antbpio S.Christovao sranaia imbo inhas antos ?Among other manufactures are butter and cheese, canned fruits and vegetables, glass and earthenware, printing and wrapping paper, furniture, matches, hats, clothing, pharmaceutical products, soaps and -

**p**erfumery, ice, artificial drinks, cigars and cigarettes, fireworks anc candles.The proud minister had been resisted

**p**in his plans of reform at home by the Jesuits, and, determining to attack the power of the order, first deprived them of all temporal power in the state of Maranhao and Para.According to these men, even though rationality did not exist in any individual, its existence in nature would still remain intact " (Cousin, Introduction, &c.,

**p**. cxx.).During the long reign of Sigismund (1387-1437) Hungary was brought face to face with the Turkish peril in its most threatening shape, and all the efforts of the king were directed Turkish Turks crossed the Hellespont from Asia Minor and

**p**began that career of conquest which made them the terror of Europe for the next three centuries.**P**g g the emperor Maximilian and Sigismund of Poland, might be dispensed with.In 1823, when the reactionary powers were meditating joint action to suppress the - revolution in Spain, the government without consultin

**P**?On the 17th of February 1867 a responsible inde pendent ministry was formed under Count Gyula '

**p**y y ' of 1867.**P**g**P****P**ment a royal autograph letter stating the reasons which had actuated the king in taking this course, and giving as the task of the new ministry the continuance of negotiations with the Coalition on the basis of the exclusion of the language question.Y, 9

**P**?, PP ment of a Coalition cabinet 2 under Dr Sandor Wekerle was announced, the world was taken completely by surprise.Thus, while of German periodicals ap

**p**earing in Hungary there were in 1871 only 85, they increased in 1880 to 114, in 1885 to 141; and they were, at the beginning of 1895, still 128, in spite of the constant spread of that process of Magyarization which has, since 1880, considerably changed the linguistic habits of the people of Hungary.For multiplication, for instance, we have the statement that, if

**P**and Q are two quantities, containing respectively**p**and q of a particular unit, then**p**X Q = q X**P**; or the more abstract statement that**p**X q= q X**p**.In n = a

**P**, a is the root or base,**p**is the index or logarithm, and n is the power or antilogarithm.But a

**P**is sometimes incorrectly described as " a to the power**p**"; the power being thus confused with the index or logarithm.It should be observed that, by analogy with the definition of a fraction, a

**P**l q mean (al/q)**P**, not (aP)llq.Thus

**P**= kQ+R, where k is an integer.Having obtained R, which is less than Q, we now repeat with Q and R the process that we adopted with

**P**and Q; i.e.Hence u is the greatest common measure of

**p**and q.In order that a monomial containing a m as a factor may be divisible by a monomial containing a

**p**as a factor, it is necessary that**p**should be not greater than m.Thus from

**P**+Q - R+S=T we deduce**P**+(Q - R+S)=**P**+(T -**P**).Thus, if we have an equation

**P**=Q, where**P**and Q are numbers involving fractions, we can clear of fractions, not by multiplying**P**and Q by a number m, but by applying the equal multiples**P**and Q to a number m as unit.This process consists in proving that a property involving

**p**is true when**p**is any positive integer by proving (I) that it is true when**p**= 1, and (2) that if it is true when**p**=n, where n is any positive integer, then it is true when**p**= n+ I.Then the assumption is that, whatever positive integral value n may have, n [

**P**] is divisible by**p**!.But, by hypothesis, n 1P1 is divisible by

**p**!.Therefore, if (n-i) 1P+11 is divisible by (

**p**+I)!, 'n 1P+11 is divisible b y (**p**+i) !.These constructed symbols may be called positive and negative coefficients; or a symbol such as (-

**p**) may be called a negative number, in the same way that we call 3 a fractional number.The result of the extension is that the number or quantity represented by any symbol, such as

**P**, may be either positive or negative.We therefore define algebraical division by means of algebraical multiplication, and say that, if

**P**and M are multinomials, the statement "**P**/M = Q " means that Q is a multinomial such that MQ (or QM) and**P**are identical.It can be proved by geometry that (aA-H3B) +yC = aA+(aB+- y C) = (a + 1 3+ 7)

**P**, where**P**is in fact the centroid of masses a, 13, y placed at A, B, C respectively.We imagine a wave-front divided o x Q into elementary rings or zones - often named after Huygens, but better after Fresnelby spheres described round

**P**(the point at which the aggregate effect is to be estimated), the first sphere, touching the plane at 0, with a radius equal to PO, and the succeeding spheres with radii increasing at each step by IX.The component vibrations at

**P**due to the successive zones are thus nearly equal in amplitude and opposite in phase (the phase of each corresponding to that of the infinitesimal circle midway between the boundaries), and the series which we have to sum is one in which the terms are alternately opposite in sign and, while at first nearly constant in numerical magnitude, gradually diminish to zero.Now as to the phase of the secondary wave, it might appear natural to suppose that it starts from any point Q with the phase of the primary wave, so that on arrival at

**P**, it is retarded by the amount corresponding to QP. But a little consideration will prove that in that case the series of secondary waves could not reconstitute the primary wave.Further, it is evident that account must be taken of the variation of phase in estimating the magnitude of the effect at

**P**of the first zone.If the primary wave at 0 be cos kat, the effect of the secondary wave proceeding from the element dS at Q is dS 1 dS -

**p**cos k(at -**p**+ 4 A) = - -- sin k(at -**p**).If a formal proof be desired, it may be obtained by introducing into the integral a factor such as

**P**, in which h is ultimately made to diminish without limit.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.In the applications with which we are concerned, t, n are very small quantities; and we may take

**P**= x yn - At the same time dS may be identified with dxdy, and in the de nominator**p**may be treated as constant and equal to f.In these expressions we are to replace

**p**by ks/f, or rather, since the diffraction pattern is symmetrical, by kr/f, where r is the distance of any point in the focal plane from the centre of the system.The greatest brightness is at the centre, where dC = 27rp d

**p**, C = 7rR2.Now by (17), (18) z 1 J1(z) =Jo(z)- (z); where

**P**increases.When the interval is very small the discrepancy, though mathematically existent, produces no practical effect, and the illumination at B due to

**P**is as important as that due to A, the intensities of the two luminous sources being supposed equal.Under these conditions it is clear that A and

**P**are not separated in the image.This is necessarily a question of degree; but it does not require detailed calculations in order to show that the discrepancy first becomes conspicuous when the phases corresponding to the various secondary waves which travel from

**P**to B range over a complete period.The illumination at B due to

**P**then becomes comparatively small, indeed for some forms of aperture evanescent.The extreme discrepancy is that between the waves which travel through the outermost parts of the object-glass at L and L'; so that if we adopt the above standard of resolution, the question is where must

**P**be situated in order that the relative retardation of the rays PL and PL' may on their arrival at B amount to a wave-length (X).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.In spite of any inequality between

**p**and**p**', the definition will be good to this order of approximation, provided a and y vanish.A similar expression can be found for Q'

**P**- Q"A; and thus, if Q' A =v, Q' AO = where v =a cos (0", we get - - -AQ' = a sin w (sin 4 -sink") - - 8a sin 4 w(sin cktan 4 + sin 'tan cl)').In the present application 4' is not necessarily equal to; but if

**P**correspond to a line upon the grating, the difference of retardations for consecutive positions of**P**, so far as expressed by the term of the first order, will be equal to mX (m integral), and therefore without influence, provided v (sin 0-sin0') = nzX (11), where a denotes the constant interval between the planes containing the lines.If we write

**p**= 27rR/A (6), we must regard**p**as a function of f, and we may take with sufficient approximation under any ordinary circumstances where**p**' denotes the value of**p**at 0, and is a constant, which is positive when the retarding plate is held at the side on which the blue of the spectrum is seen.Only in this case can cos {

**p**' +(m- -27th/Af) f } retain the constant value - I throughout the integration, and then only when and a = 27Th/A f (8) cos**p**'=- 1 ..The linear width of the band (e) is the increment of which alters

**p**by 27r, so that e =27r /tr.But against this explanation of the heading ry;

**p**' 2 there is an almost insuperable objection; for, since both the first and second books contain psalms with this heading, it is clear that the " Chief Musician's - or Director's - Psalter " must have been in existence before either of these books; in which case, apart from the difficulty of the antiquity which we should be compelled to assign to this earliest Psalter, it is impossible to understand on what principle the first book of Psalms was formed.Now we have seen that the 5 prefixed to n-'

**p**'**p**cannot refer to authorship; we seem therefore shut up to one of two alternatives, either the psalms inscribed 's?It may be safely said of all those living things which are large enough to enable us to trust the evidence of microscopes, that they are heterogeneous optically, and that their different parts, and especially the surface layer, as Life and contrasted with the interior, differh sicall and organiza-

**P**Y ?But many of the simpler

**P**Y ler**P**forms of life may undergo desiccation to such an extent as to arrest their vital manifestations and convert them into the semblance of not-living matter, and yet remain potentially alive.He has no claim to be regarded as a genius; but, as SainteBeuve has said, he well deserves a place "da

**p**s la classe des esprits infiniment distingues" - distinguished, however, it ought to be added by intelligence rather than by intellect, and less by the power of saying much than by the power of saying a little well.The present London residence of the sovereign is Buckingham Palace, on the west side of St

**p**James's Park, with beautiful gardens behind it.**P**of the city was its special property, and it extended as far as the limits of the territorium of the nearest Roman city or as near thereto as the natural boundaries."Except in a few instances these were long ago superseded by ron-wire ropes, which in turn have

**p**been replaced by steel because of its greater strength.While the mine workings are small the overlying rocks support themselves of and the full pressure does not come upon the mine Caving

**p**i llars.The fighting spirit of the people of for iveness and endurance, upheld Guru Nanak's g

**p**was roused and satisfied by the spiritual and military leader.His elder brother Ram Rai was passed over

**p**was put to death for refusal to embrace Islam b.Let

**P**, Q denote the normal thrust across the sides bc, ca, and R the normal thrust across the base ab.Then, since these three forces maintain equilibrium, and R makes equal angles with

**P**and Q, therefore**P**and Q must be equal.But the faces bc, ca, over which

**P**and Q act, are also equal, so that the pressure on each face is equal.If a thrust

**P**lb is applied to one piston of area A ft.For if the liquid of density a rises to the height h and of density

**p**to the height k, and po denotes the atmospheric pressure, the pressure in the liquid at the level of the surface of separation will be ah+Po and pk +po, and these being equal we have Uh = pk.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.These equations can be made to represent the state of convective equilibrium of the atmosphere, depending on the gas-equation

**p**= pk =RA (6) where 0 denotes the absolute temperature; and then d9 d**p**R dz - dz (**p**) n+ 1' so that the temperature-gradient deldz is constant, as in convective equilibrium in (I I).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.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?Then dp/dz=kdp/dz =

**P**, = Poe ik,**p**- po= kpo(ez Ik -1); (16) and if the liquid was incompressible, the depth at pressure**p**would be (**p**- po) 1po, so that the lowering of the surface due to compression is ke h I k -k -z= 1z 2 /k, when k is large.For a homogeneous liquid at rest under gravity,

**p**is proportional to the depth below the surface, i.e.Suppose

**P**tons is moved c ft.The varying direction of the inclining couple Pc may be realized by swinging the weight

**P**from a crane on the ship, in a circle of radius c. But if the weight**P**was lowered on the ship from a crane on shore, the vessel would sink bodily a distance**P**/wA if**P**was deposited over F; but deposited anywhere else, say over Q on the water-line area, the ship would turn about a line the antipolar of Q with respect to the confocal ellipse, parallel to FF', at a distance FK from F FK= (k2-hV/A)/FQ sin QFF' (2) through an angle 0 or a slope of one in m, given by**P**sin B= m wA FK - W'Ak 2V hV FQ sin QFF', (3) where k denotes the radius of gyration about FF' of the water-line area.I n a straight uniform current of fluid of density

**p**, flowing with velocity q, the flow in units of mass per second across a plane area A, placed in the current with the normal of the plane making an angle 0 with the velocity, is oAq cos 0, the product of the density**p**, the area A, and q cos 0 the component velocity normal to the plane.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.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.Thus if d,/ is the increase of 4, due to a displacement from

**P**to**P**', and k is the component of velocity normal to PP', the flow across PP' is d4 = k.PP'; and taking PP' parallel to Ox, d,, = vdx; and similarly d/ ' = -udy with PP' parallel to Oy; and generally d4,/ds is the velocity across ds, in a direction turned through a right angle forward, against the clock.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.Motion symmetrical about an Axis.-When the motion of a liquid is the same for any plane passing through Ox, and lies in the plane, a function ' can be found analogous to that employed in plane motion, such that the flux across the surface generated by the revolution of any curve AP from A to

**P**is the same, and represented by 2s-4 -11'o); and, as before, if d is the increase in due to a displacement of**P**to**P**', then k the component of velocity normal to the surface swept out by PP' is such that 274=2.7ryk.PP'; and taking PP' parallel to Oy and Ox, u= -d/ydy, v=dl,t'/ydx, (I) and 1P is called after the inventor, " Stokes's stream or current function," as it is constant along a stream line (Trans.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.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.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?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'.The image of a source of strength

**p**at S outside a sphere of radius a is a source of strength pa/f at H, where 'OS' =f, OH =a2/f, and a line sink reaching from the image H to the centre 0 of line strength - A la; this combination will be found to produce no flow across the surface of the sphere.Taking Ox along OS, the Stokes' function at

**P**for the source S is**p**cos PSx, and of the source H and line sink OH is**p**(a/f) cos PHx and - (**p**/a) (PO - PH); so that =**p**(cos PSx+f cos PHx PO a PH), (q) and Ili = -**p**, a constant, over the surface of the sphere, so that there is no flow across.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 =..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.The cuneiform system of writing Semitic. w as still in process of growth when it was borrowed influence

**p**g and adapted by the new comers, and the Semitic Babylonian language was profoundly influenced by the older language of the country, borrowing its words and even its grammatical usages.**P**Y administrative abilities of his father, and the success of his reign was not commensurate with the vanity of the ruler.His successor was Nabo oPP Y

**P**lassar, between whom and the last king of Assyria war broke out.In 1347 Florence was again stricken with famine, followed the next year by the most terrible plague it had ever experienced, which carried off three-fifths of the population (according to now threatened Florence in the person of Castruccio

**p**Villani).This g Y

**P**Y PP Y infamous law, however, aroused so much opposition that some of the very men who had proposed it assembled in secret to discuss its abolition, and a quarrel between the Albizzi and the Ricci having weakened the parte, a balia of 56 was agreed upon.If by this process a more perfect defecation and purification of the juice is obtained, it will no doubt be highly beneficial to the cane

**p**lanter, though no great economy in lime can be effected, because but very little is used in a cane factory in comparison with the amount used in a beet factory.Let PQ be the ordinate of the point

**P**FIG.Outline of the ventral surface to show the external apertures and nervous system; a, rosette-organ; b, uterine pore; c, terminal sucker; e, vaginal pore; g, male gonopore; n, o,

**p**, nervous system.Where the amount of phosphoric acid (

**P**2 0,) is less than.The paring and burning of land, although formerly practised as an ordinary means of improving the texture and fertility of arable fields, can now only be looked upon as a practice

**p**to be adopted for the purpose of bringing rapidly into cultivation very foul leys or, land covered with a coarse turf.Palgrave made his adventurous journey through Nejd, and published the remarkable narrative which has taken itslace as the classic of Arabian

**P**Nejd.On his death in 1897 his nephew Abdul-Aziz, son of the murdered amir Matab, succeeded; during his reign a new element has been introduced into Nejd politics by the rising importance of Kuwet (Koweit) and the attempts R t g

**P**()**P**history.At the close of the month he resigned his

**p**ost on being elected, in spite of his youth, a deputy to the Convention by the department of Seine-et-Oise, and he began his legislative career by defending the conduct of the Commune during the massacres.Returning to favour in 1899, he was promoted to the Legation at Tokio, where, however, under the influence of German reports concerning the Japanese army - and es

**p**ecially its artillery - he misjudged Japan's advent as a Great Power.If M 1, M2, and

**P**1,**P**2 be the molecular weights and vapour pressures of the components A and B, then the ratio of A to B in the distillate is M 1**P**1 /M 2**P**2.**P**1 greater than P2, if the molecular weight of A be much less than that of B, then it is obvious that the ratio M 1**P**1 /M 2**P**2 need not be very great, and hence the less volatile liquid B would come over in fair amount.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.A, Dorsal view showing the nervous system and digestive system; a, mouth; b, pharynx; c, d, e, gut; E, post-genital union of two limbs of gut; f, excretory pore; g, vaginal pore; h, j, k, brain and nerves; 1, dorsal nerves; m, ventral nerves; n, adoral sucker; o, posterior sucker;

**p**, hooks on posterior sucker; r, vitello-intestinal duct.S' 3' S'

**p**called voluntary loans was abolished, and replaced by a tax of ro% (la decima) on all real property.Bribery was the first weapon employed, and a car- y

**p**dinal's hat was held out as a bait.So long as King Charles remained in Italy Alexander's concern for his own safety

**p**revented vigorous measures against the friar.His pulpit in the duomo was defiled, an ass's skin spread over the cushion and shar nails fixed in the board Bxcorn-,

**p**mun.cated.In other Entoprocta the buds on a horseshoe retain a high degree of individuality, a shaped l o

**p**h o - thread-like stolon giving off the cylindrical phore; stalks, each of which dilates at its end i, Ectocyst; into the body of a zooid.Vast

**p**PRINTED If ouds of dust and stones, blown out of the crater and funnel of ie volcano, were hurled into the air and carried for hundreds miles, the finer particles falling to the earth even.The Reason is the superior andre onderant element which settles the direction

**p****p**in which all the other faculties shall expand.They are said, though it is not easy to

**p**believe, to have been elaborated by way of Utopia.Sir Robert Peel became r i me minister, and made the member for Newark

**p**vice-president of the Board of Trade.Having been returned again for the university of Oxford, he entered on the active duties of a great office for which he was pre-eminently g

**P**Y House of government in power, but on the 18th of October the commons.He was now sixty-four, and his life had been a continuous experience of exhausting Temporary

**p**g retirement.As for his ivory-white, it distinctly surpasses the Chinese Ming Chen-yao in every quality except an indescribable intimacy of glaze and

**p**&e which probably can never be obtained by either Japanese or European methods.The platinum strips are exceedingly minute, being in some cases only 2 5

**p**in.The Priestly Writer in the Pentateuch also a

**p**pears to be acquainted with this doctrine; it is the first of four ages which begins with the Creation and ends with the Deluge.The small eyes are some- T y

**p**h l o**p**s bra- times covered by transparent shields.In such a diagram, a point

**P**defines a particular mixture, both as to percentage, composition and temperature; a vertical line through**P**corresponds to the mixture at all possible temperatures, the point Q being its freezing-point.His genius still works under forms prescribed by Greek art, and under the disadvantage of having a

**p**ractical and utilitarian aim imposed on it.In the beginning of March 1921, direct negotiation between Poland and Lithuania under the aus

**p**ices of the League of Nations, to be followed by arbitration on unsettled points, was proposed in lieu of the plebiscite and agreed to by all parties.Thus if Q is the surface density, S the thickness of the shell at any point, and

**p**the assumed volume density of the matter of the shell, we have v =Abp. Then the quantity of electricity on any element of surface dS is A times the mass of the corresponding element of the shell; and if Q is the whole quantity of electricity on the ellipsoid, Q =A times the whole mass of the shell.This mass is equal to 47rabcp,u; therefore Q = A47rabcp s and b =pp, where

**p**is the length of the perpendicular let fall from the centre of the ellipsoid on the tangent plane.Let this rectangular prism be supposed to be wholly filled up with electricity of density

**p**; then the total quantity in it is**p**dx dy dz.It defines the condition which must be fulfilled by the potential at any and every point in an electric field, through which

**p**is finite and the electric force continuous.It may be looked upon as an equation to determine

**p**when V is given or vice versa.Then bearing in mind that a= (I/4x1-)dV/dn, and

**p**=-(1/4xr)VV, we have finally E 2 c/v=2 f f v.-dS+ 2J J J Vp dv.Then the potential at any point

**P**in this ideal plane PO is equal to q/AP-q/BP=0, whilst the resultant force at**P**due to the two point charges is 2gAO/AP 3, and is parallel to AB or normal to PO.The isothermals are approximately equilateral hyperbolas (pv= constant), with the axes of

**p**and v for asymptotes, for a gas or unsaturated vapour, but coincide with the isopiestics for a saturated vapour in presence of its liquid.Substituting these symbols in the expression for the area, the relation becomes H=o(

**p**' -**p**„)(v„_v')/(o' - o").In the notation of the calculus the relations become - dH/dp (0 const) = odv /do (

**p**const) (4) dH/dv (0 const) =odp/do (v const) The negative sign is prefixed to dH/dp because absorption of heat +dH corresponds to diminution of pressure - dp. The utility of these relations results from the circumstance that the pressure and expansion co efficients are familiar and easily measured, whereas the latent heat of expansion is difficult to determine.Finally, the substance is reconverted into the first state at the temperature 0", completing the cycle by the abstraction of a quantity of heat By the application of the first law, the difference of the quantities of heat absorbed and evolved in the cycle must be equal to the work represented by the area of the cycle, which is equal to (

**p**' -**p**") (v" - v') in the limit when the difference of pressure is small.Dividing by (0' - e"), and writing dp/do and dL/do for the limiting values of !the ratios (

**p**' -**p**")/(o' - o") and (L' - L")/(o' - o"), we obtain the important relations s' - s"+dL/do= (v" - v')dp/do=L/o,..For a finite change it is necessary to represent the path by a series of small steps, which is the graphic equivalent of integration along the path represented by the given relation between v and 0, or

**p**and 0.The change of energy at constant volume is simply sdo, the change at constant temperature is (odp/de -

**p**)dv, which may be written dE/de (v const) =s, dE/dv (0 const) =odp/do -**p**.A similar expression for the variation of the specific heat S at constant pressure is obtained from the second expression in (8), by taking

**p**and 0 as independent variables; but it follows more directly from a consideration of the variation of the function (E+pv).To find the total heat of a substance in any given state defined by the values of

**p**and 0, starting from any convenient zero of temperature, it is sufficient to measure the total heat required to raise the substance to the final temperature under a constant pressure equal to**p**. For instance, in the boiler of a steam engine the feed water is pumped into the boiler against the final pressure of the steam, and is heated under this constant pressure up to the temperature of the steam.Observing that F is a function of the co-ordinates expressing the state of the substance, we obtain for the variation of S with pressure at constant temperature, dS/dp (0 const) '=' 2 F/dedp =-0d 2 v/d0 2 (

**p**const) (12) If the heat supplied to a substance which is expanding reversibly and doing external work, pdv, is equal to the external work done, the intrinsic energy, E, remains constant.The equation to these lines in terms of v and 0 is obtained by integrating dE=sd0+(Odp/de -

**p**)dv = o .The equation to the lines of constant total heat is found in terms of

**p**and 0 by putting dF=o and integrating (it).The isothermal elasticity - v(dp/dv) is equal to the pressure

**p**. The adiabatic elasticity is equal to y**p**, where -y is the ratio S/s of the specific heats.It is found by experiment that the change of pv with pressure at moderate pressures is nearly proportional to the change of

**p**, in other words that the coefficient d(pv)/dp is to a first approximation a function of the temperature only.If we consider any short length of the stream bounded by two imaginary cross-sections A and B on either side of the plug, unit mass of the fluid in passing A has work,

**p**'v', done on it by the fluid behind and carries its energy, E'+ U', with it into the space AB, where U' is the kinetic energy of flow.In passing B it does work,

**p**"v", on the fluid in front, and carries its energy, E"+ U", with it out of the space AB.We have therefore, by equation, (11), Sd0 = (Odv/d0 - v) d

**p**,.The characteristic equation of the fluid must then be of the form v/0=f(

**p**), where f(**p**) is any arbitrary function of**p**. If the fluid is a gas also obeying Boyle's law, pv = f (0), then it must be an ideal gas.Putting d0/dp=A/0 2 in equation (15), and integrating on the assumption that the small variations of S could be neglected over the range of the experiment, they found a solution of the type, v/0 =f(

**p**) - SA /30 3, in which f(**p**) is an arbitrary function of**p**. Assuming that the gas should approximate indefinitely to the ideal state pv = R0 at high temperatures, they put f(**p**)=Rip, which gives a characteristic equation of the form v= Re/**p**- SA /30 2 .Other favourite types' of equation for approximate work are (I)

**p**=RO/v±f(v), which makes**p**a linear function of 0 at constant volume, as in van der Waal's equation; (2) v=RO/**p**+f(**p**), which makes v a linear function of 0 at constant pressure.Denoting by So, so, these constant limiting values at

**p**=o, we may obtain the values at any pressure by integrating the expressions (27) and (28) from co to v and from o to**p**respectively.If the tube is a perfect non-conductor, and if there are no eddies or frictional dissipation, the state of the substance at any point of the tube as to E,

**p**, and v, is represented by the adiabatic or isentropic path, dE= -pdv.Since dJ=-4d9-pdv, we have also the relations dJ'/dv' = -

**p**= dJ"/dv", at constant temperature.This may be interpreted as the equation of the border curve giving the relation between

**p**and 0, but is more easily obtained by considering the equilibrium at constant pressure instead of constant volume.Assuming the function G to be expressed in terms of

**p**and 0, this condition represents the relation between**p**and 0 corresponding to equilibrium between the two states, which is the solution of the relation (v" - v')dp/dO=L/D, (5).The direct integration of this equation requires that L and v" - v' should be known as functions of

**p**and 0, and cannot generally be performed.Thus in Greece there are eleven archbishops

**p**to thirteen bishops, the archbishop of Athens alone being metropolitan; in Cyprus, where there are four bishops and only one archbishop, all five are of metropolitan rank.It is the latest writer (

**P**) who mentions Abram (the original form of the name), Nahor and Haran, sons of Terah, at the close of a genealogy of the sons of Shem, which includes among its members Eber the eponym of the Hebrews.For the completion of the history of Abraham the compiler of Genesis has used

**P**'s narrative.In this alphabet the Greek letter

**p**(or rather a very similar letter with the loop a little lower down) is used to represent sh, and there are some peculiarities in the use of o apparently connected with the expression of the sounds h and w.The reason for this is readily seen; if a mass M of any gas occupies a volume V at a temperature T (on the absolute scale) and a pressure

**P**, then its absolute density under these conditions is O = M/V; if now the temperature and pressure be changed to l and**P**,, the volume V l under these conditions is VPT/PIT1, and the absolute density is MP,T/VPT I.From the bottom there leads

**P**another fine tube, bent upwards, and then at right angles so as to be at the same level as the capillary branch.The tube V, on the contrary, is attached to the cradle, and merely forms a support for the finder Q, the handles at f and

**p**, and the moving ring**P**. The latter gives quick motion in position angle; the handles at**p**clamp and give slow motion in position angle, those at f clamp and give slow motion in right ascension and declination.The reverse motion of

**P**automatically moves the paper ribbon forward, ready to - 20 receive the next impression.The only connexion between the type-wheel and the screwhead S is by the pin

**p**(which is screwed into S), the cylindrical end of which acts in a slot cut in the typewheel.The prism

**p**is fitted accurately into brass slides (care has to be taken in the construction to place the prism so that an object in the centre of the field will so remain when the eye-piece is rotated in its adapter).The magnifying power is varied by changing the lens a for another in which

**p**has a different value.If

**p**" is the refracting angle of the prism, and n the magnifying power of the eye-piece, then**p**"ln will be the distance observed.Of the rest,

**P**and Q are imperfect.For by transliterating Ka7vap Nepc'ev into Hebrew 1113 10,

**P**and adding together the sums denoted by the Hebrew letters we obtain the number 666.In the tropical and subtropical belts of the Atlantic and Indian Oceans south of the equator the salinity diminishes rapidly from the surface downwards, and at 500 fathoms reaches a minimum of 34.3 or 34.4

**p**e r mille; after that it increases again to 800 fathoms, where it is almost 34.7 or 34.8, and this salinity holds good to the bottom, even to the greatest depths, as was first shown by the " Gauss " and afterwards by the " Planet " between Durban and Ceylon.The shaft is lined with a cylinder of wrought iron, within which a tubular chamber, provided with doors above and below, known as an

**P**g air-lock, is fitted by a telescopic joint, which is tightly sinkin packed so as to close the top of the shaft air-tight.The surface arrangements of a modern deep colliery are of considerable extent and complexity, the central feature being the head gear or pit frame carrying the guide pulleys Surface which lead the winding roes from the axis of the it arrange= g

**P****P**to the drum.If

**P**represent the average value of the component of a force in the direction of the displacement, s, of its point of application, the product Ps measures the work done during the displacement.The apparatus used to measure

**P**or T is the dynamometer.Apparatus is added to some dynamometers by means of which a curve showing the variations of

**P**on a distance base is drawn automatically, the area of the diagram representing the work done; with others, integrating apparatus is combined, from which the work done during a given interval may be read off directly.This has a slotted end, engaged by a pin

**P**fixed to the framing, and it will be seen that its action is to slacken the band if the load tends to rise and to tighten it in the contrary case.The external forces holding the brake from turning are W, distant R from the axis, and the reaction, W 1 say, of the lever against the fixed pin

**P**, distant R1 from the axis.When the wheel is turning in the direction indicated, the forces holding the band still are W, and

**p**, the observed pull on the spring balance.Both these forces usually act at the same radius R, the distance from the axis to the centre line of the rope, in which case the torque T is (W-

**p**)R, and consequently the brake horse-power is (W -**p**)RX21rN, When µ 33,000 changes the weight W rises or falls against the action of the spring balance until a stable condition of running is obtained.The ratio

**p**is given by e"` e, where e= 2.718; µ is the coefficient of friction and 0 the angle, measured in radians,, subtended by the arc of contact between the rope and the wheel.The ratio W/

**p**increases very rapidly as 0 is increased,, and therefore, by making 0 sufficiently large,**p**may conveniently be made a small fraction of W, thereby rendering errors of observation of the spring balance negligible.When a belt, in which the maximum and minimum tensions are respectively

**P**and**p**lb, drives a pulley, the torque exerted FIG.**P**and**p**may be measured directly by leading the belt round two freely hanging guide pulleys, one in the tight, the other in the slack part of the belt, and adjusting loads on them until a stable condition of running is obtained.If a force Q acting at R maintains equilibrium, QR/4 = (

**P**-**p**)r =T.The force Q, usually measured by a spring, required to maintain the beam in its central position is proportional to (

**P**-**p**).If the angle 0 1 =0 2 =120 0, Q = (

**P**-**p**) neglecting friction.When a shaft is driven by means of gearing the driving torque is measured by the product of the resultant pressure

**P**acting between the wheel teeth and the radius of the pitch circle of the wheel fixed to the shaft.This is equivalent to a steady pressure

**p**i per unit area where +0 pi - zfff v J 1 (h3m3/ir3)e hm(u2+v2+w2)mu2dudvdw.This equation is found experimentally to be capable of representing the relation between

**p**, v, and T over large ranges of values.The equation of energy is dQ=dE+pdv, (17) expressing that the total energy dQ is used partly in increasing the internal energy of the gas, and partly in expanding the gas against the pressure

**p**. If we take**p**= RNT/v from equation (14) and substitute for E from equation (16), this last equation becomes dQ 2 (n +3)RNdT +RNTdv (18) which may be taken as the general equation of calorimetry, for a gas which accurately obeys equation (14).The Cathars even held it necessary, in case a bishop fell into mortal sin, to repeat his ba

**p**tisms and ordinations, for they had been vitiated by his sins.The laws of the Carolingian empire provided that one excom municated by the Church who did not make his peace

**p**within a year and a day should be outlawed, and this general principle was not lost sight of.Y

**P**Here he began to denounce the abuses in the Church, as well as the traffic in mercenaries which had so long been a blot upon his country's honour.As might have been anticipated, this caused no break in the policy of the English king and his parliament, and a series of famous acts passed in the year 1534 completed and confirmed the independence of the Church of England, which, except during five years under Queen Mary,

**p**g Y Q Y?Contemporaneously with the Wars of Religion in France a long and terrible struggle between the king of Spain and his Dutch and Belgian provinces had resulted in the formation of a Protestant state - the United Nether- United lands, which was destined tola an important role play

**p**in the history of the Reformed religion.Open both to German and French influences, the Netherlands had been the scene of the first executions of Lutherans; they had been a centre of Anabaptist agitation; but Calth y

**P**?A philosopher," as Gibbon long ago pointed out, _ who asks from what articles of faith above and against reason the early Reformers enfranchised their followers of

**P**will be surprised at their timidity rather than scandal Y ized by their freedom.Ellis, l ater expanded, and in the process somewhat weakened, into his

**P**liritan Age and Rule in the Colony of the Massachusetts Bay, 1629-168 5 (Boston, 1888; 3rd ed., 1891).Whether with Payne it is assumed that in some remote time a The speechless anthropoid passed over a land bride now American

**P****P****P**g ?The only basis of division practicable is Y

**P**language, which must be kept separate in the mind from the others.If a .JP solid circle be fixed in any one position and a tube be pivoted on its centre so as to move; and if the line C D be drawn upon the circle pointing towards any object Q in the heavens which lies in the plane of the circle, by turn ing the tube A B towards any other object

**P**in the plane of the circle, the angle B 0 D will be the angle subtended by the two objects**P**and Q at the eye.Then MA'B'N is a right trapezium, whose area is equal to that of Cabd; and it is related to the latter in such a way that, if any two lines parallel to AC and BD meet AB, CD, MN, A'B', in E, G,

**P**, E', and F, H, Q, F', respectively, the area of the piece PE'F'Q of the right trapezium 'B.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 the prismoid is divided into a pyramid with vertex

**P**and base ABCD ..., and a series of tetrahedra, such as PABa or PAab.This would involve

**p**and q; but, for our purposes, the data are the sides pa+q and pb+q and the base b - a, and the expression of the integral in terms of these data would require certain eliminations.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.Simpson's (first) formula, for instance, holds for f = I, and is obtained by taking

**p**= i and ignoring differences after 52up.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.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+..Combining this with the first equation, we obtain the values of

**P**, Q, R, ..The general expression, if

**p**, q, r,..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'?The parts of the range of moulds are brought tightly together and held in position by the bars 0 and the screw

**P**, and when one mould is filled the carrier is moved forward on its rails by wheels worked by a handle also shown in the figure.Gioja's latest work Filosofia della statistica (2 vols., 1826;

**p**vols., 1829-1830) contains in brief compass the essence of his ideas on human life, and affords the clearest insight into his aim and method in philosophy both theoretical and practical.As stated under Phosphorus, phosphoric oxide,

**P**2 0 5, combines with water in three proportions to form H 2 O**P**2 0 5 or HP03, metaphosphoric acid; 2H 2 O**P**2 0 5 or H4P207, pyrophosphoric acid; and 3H 2 O**P**2 0 5 or H 3 PO 4, orthophosphoric or ordinary phosphoric acid.Pyrophosphoric acid, 'H' 4

**P**2 0 7, is a tetrabasic acid which may be regarded as derived by eliminating a molecule of water between two molecules of ordinary phosphoric acid; its constitution may therefore be written (HO) 2 0P O PO(OH) 2.Being a tetrabasic acid it can form four classes of salts; for example, the four solium salts Na 4

**P**2 0 7, Na3HP207, Na 2 H 2**P**2 0 7, NaH 3**P**2 0 7 are known.The most important is the normal salt, Na 4

**P**2 0 7, which is readily obtained by heating disodium orthophosphate, Na 2 HPO 4.To prevent leakage over the surface of the insulating covering of the wire which projects above the surface of the water, it is necessary to employ a " guard wire

**P**, which consists of a piece of fine copper wire, twisted round the extremity of the insulated wire and connected to the battery.In addition there is a pressure between the layers of the medium, and if this pressure in the undisturbed parts of the medium is

**P**, momentum**P**per second is being transferred from right to left across each square centimetre.If the velocity of a particle at A relative to the undisturbed parts is u from left to right, the velocity of the matter moving out at A is U - u, and the momentum carried out by the moving matter is

**p**(U - u) 2.Equating this to the momentum entering at B and subtracting

**P**' from each X+W+**p**(U - u)2 =poU 2.He supposed that in air Boyle's law holds in the extensions and compressions, or that

**p**= kp, whence dp/dp = k =**p**/**p**. His value of the velocity in air is therefore U = iJ (**p**ip.) (Newton's formula).That is to say, instead of using Boyle's law, which supposes that the pressure changes so exceedingly slowly that conduction keeps the temperature constant, we must use the adiabatic relation

**p**= kpy, whence d**p**/d**p**= y k**p**Y 1= yp/**p**, and U = (yp/**p**) [Laplace's formula].For two different gases with the same value of y, but with densities at the same pressure and temperature respectively

**p**i and p2, we should have U1/U2 =1 1 (P2/P1), (Io) another result confirmed by observation.When the value of dyldx is not very small E is no longer constant, but is rather greater in compression and rather less in extension than -yP. This can be seen by considering that the relation between

**p**and is given by a curve and not by a straight line.If

**P**is the undisturbed pressure and**P**+w the pressure at AB, the momentum entering through AB per second isJ01(**P**+w-+pu2)dt.Buti o Pdt =

**P**is the normal pressure, and as we only wish to find the excess we may leave this out of account.Then putting (dy/dx) 2 = (u/U) 2, we have y +I I fU

**p**= 2.If the train of waves is reflected, the value of

**p**at AB will be the sum of the values for the two trains, and will, on the average, be doubled.Comparing the velocities of sound U i and U2 in two different gases with densities and at the same temperature and pressure, and with ratios of specific heats 'yl, 72, theory gives Ui/U2 = 1/ {71

**p**2/72**p**i }.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 d is measured for two gases in succession for the same frequency N, we have 72

**p**2P1 d22 71**p**i**p**s d12' where the suffixes denote the gases to which the quantities relate.From the temperature,

**P**2 /**p**2 was known, and 72 / 71 could then be found.Take a point

**P**in the disturbed part, and a point Q which the disturbance has not yet reached.But the tension at

**P**is T, parallel to the tangent, and T sin 4 parallel to PM, and through this - T sin is the momentum passing out at**P**per second.If

**p**is the density at A, and w the cross-section, then the momentum carried past A is pc,(U - u) 2.Then we have Ya,ody/dx +

**p**& (U - u) 2 =**p**o&,oU 2.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.The mass of matter moving through A per second is pwa 2 U, where a is the radius of the wire and

**p**is its density.Let us suppose that with constant excess of pressure,

**p**, in the wind-chest, the amplitude produced is proportional to the pressure, so that the two tones issuing may be represented by pa sin 27-nit and pb sin 21rn 2 t.To the Guard and 2nd divisions was assigned the frontal attack on the Chiuliencheng position where the Russians had about one-half of Bathe of

**P**, the Yalu.The true importance of Hebrew history had always cter

**p**y y ofcharafhe centred in the fact that this petty nation was the people work.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.Maitland describes it (Political Theories of the Middle Ages,

**p**. x.), have an essentially "statelike character."Let A be the dead load and B the live load, producing stress in a bar;

**p**=B / A the ratio of live to dead load; f i the safe working limit of stress for a bar subjected to a dead load only and f the safe working stress in any other case.Let

**p**i, P2The greatest shear at C for any position of the load occurs when the head of the train is at C. For any load

**p**between C and B will increase the reaction at B and therefore the shear at C by part of**p**, but at the same time will diminish the shear at C by the whole FIG.I

**p**- - ---??Then the moment at C due to all the loads is M = Ply1+

**P**2y2+ .Let

**P**be the cost of one pier; G the cost of the main girders for one span, erected; n the number of spans; 1 the length of one span, and L the length of the bridge between abutments.Hence the total cost of that part of the bridge which varies with the span adopted is C = (n-i)

**P**-FnaP = LP// -**P**+Lal.Many candi