# P sentence example

p

- He'd better watch his P's and Q's.
- The pseudo-coprolites of the Suffolk Crag have been estimated by Herapath to be as rich in phosphates as the true ichthyo-coprolites and saurio-coprolites of other formations, the proportion of P 2 O 5 contained varying between 12.5 and 37.25%, the average proportion, however, being 32 or 33%.
- He was born at Cordova in 1135, fled with his parents from persecution in 1148, settled at Fez in i 160, passing P g there for a Moslem, fled again to Jerusalem in 1165, and finally went to Cairo where he died in 1204.
- From Garmat Ali, where the Tigris and Euphrates at present unite,' under the title of Shattel-Arab, the river sweeps on to Basra, Ex p o yds.
- Disulphuryl chloride, S 2 O 5 C1 2, corresponding to pyrosulphuric acid, is obtained by the action of sulphur trioxide on sulphur dichloride, phosphorus oxychloride, sulphuryl chloride or dry sodium chloride: 650 3- + 2POC1 3 = P 2 O 5 + 3S 2 O 5 C1 2; S2C12+ 5503 = S 2 0 5 C1 2 + 550 2; SO 3 + SO 2 C1 2 = S 2 0 5 C1 2; 2NaC1 + 3SO 3 = S 2 0 5 C1 2 -1 Na 2 SO 4.Advertisement
- On the whole it seems most likely that, while the kernel of the Roman plebs was rural or belonged to the small towns admitted to the Roman franchise, the Attic demos, largely at least, though doubtless not wholly, arose out of the mixed settlers who had come together in the city, answering to the p rotKot of later times.
- Thus at Athens 1 its history is in its main outlines very much the same as its history at Rome up to a Y Y P certain point, while there is nothing at Athens which at all answers to the later course of things at Rome.
- In addition to this there are certain writings by his son Isidorus H€pc irpoaOuous, liuxiis; EO iy17ruca on the prophet Parchor (HapXci p); 'HBcrca.
- Y g Y P process, so manipulated as to secure an overwhelming preponderance for the wealthy, and especially the landed classes, and also for the representatives of the Russian as opposed to the subject peoples.
- As a legislative body the powers of the Council are co-ordinate with those of the Duma; in practice, however, it has seldom if ever initiated legislation.6 The Duma of the Empire or Imperial Duma (Gosudarstvennaya Duma), which forms the Lower House of the Russian parliament, consists (since the ukaz of the znd of June 1907) on the 27th of April 1906, while the name and princi p le of autocracy was jealously preserved, the word " unlimited " vanished.Advertisement
- Every industrial concern employing fifty hands or over elects one or more delegates to the electoral P ?
- By the law of the 18th of October (November i) 1905, to assist the emperor in the supreme administration a Council of Ministers (Sovyet Ministrov) was created, under a ministerresident the first a earance of a rime P, PP P minister in Russia.
- Those of them who lived on the outskirts of the pacified territory adopted a mode of life similar to that of their hereditary opponents, and constituted a peculiar class known as Cossacks, living more by flocks and The h e rds and by marauding expeditions than by a ri y g p ?'
- More recently legislation has beenassed to safeguard the lives and interests of Hours of P g Lab ur.
- P Y of risk, it has during recent years come to notice that the number of casualties among railway servants is still unduly great, and in 1899 a Royal Commission was appointed to investigate the causes of the numerous accidents, fatal and nonfatal, to railway men.Advertisement
- The points over which a train travels when directed from the main to a branch line are called " facing points " (F P), while those which it passes when running from a branch to a main line are " trailing points " (TP).
- Where, as at a double-line junction, one pair of rails crosses another pair, " diamond " crossings (p) are formed.
- Between A and B, A and C, and A and D, there may be a string of stations, p, q, r, s, &c., all receiving goods from a, b, c and d, and it would manifestly be inconvenient and wasteful of time and trouble if the trains serving those intermediate stations were made up with, say, six wagons from a to p next the engine, five from b to p at the middle, and four from c to p near the end.
- The weight W 1 carried by the part of the frame supported by the wheel (whose diameter is D) is transmitted first to the pins P 1, P2, which are fixed to the frame, and then to the spring links L 1, L2, which are jointed at their respective ends to the spring S, the centre of which rests on the axle-box.
- Let p be the mean pressure in pounds per square inch, calculated from an indicator diagram taken from a particular cylinder when the speed of the crank-shaft is n revolutions per second.Advertisement
- The plotting of the torque curve is laborious, but the average torque acting, which is all that is required for the purposes of this article, can be found quite simply, thus: - Let p be the mean effective pressure acting in one cylinder, a, the area of the cylinder, and 1, the stroke.
- Assuming that the mean pressure in the other cylinder is also p, the total work done per revolution is 4pla.
- Hence, if p is the maximum value of the mean effective pressure corresponding to about 85% of the boiler pressure,, uW = pd 2 le /D (26) is an expression giving a relation between the total weight on the coupled wheels, their diameters and the size of the cylinder.
- If p is the mean pressure at any speed the total tractive force which the engine is exerting is given by equation (25) above.
- For a stated value of the boiler pressure and the cut-off the mean pressure p is a function of the piston speed v.Advertisement
- This type has the advantage of economy in first construction, there being the minimum amount of material to be excavated, and no interference during construction with street traffic or subsurface structures; it has, however, the disadvantage of the cost of o p eration of lifts at the stations.
- The cable is slow; and unless development along new lines of com p ressed air or some sort of chemical engine takes place, electricity will monopolize the field.
- The documents underlying the Pentateuch and book of Joshua, represented by the ciphers J, E, D and P, are assumed to have been drawn up in the chronological order in which those ciphers are here set down, and the period of their composition extends from the 9th century B.C., in which the earlier portions of J were written, to the 5th century B.C., in which P finally took shape.
- It should be noted that in P (Code of Holiness) Lev.
- To a certain extent it would seem that even as Chronicles (q.v.) has passed through the hands of one who was keenly interested in the Temple service, so the other historical books have been shaped not only by the late priestly writers (symbolized in literary criticism by P), but also by rather earlier writers, also of priestly sympathies, but of " southern " or half-Edomite affinity.Advertisement
- To this age we may ascribe the literature of the Priestly writers (symbolized by P), which differs markedly from the other sources.
- It is clear that the rulers, as so P p commonly in ancient states, fulfilled priestly as well as royal functions.
- About forty years later (197) the question was discussed in a very different spirit between Victor, bisho p of Rome, and Polycrates, metropolitan of proconsular Asia.
- And as in Hebrew, the six letters b g d k p t are aspirated when immediately preceded by any vowel sound.
- Such a mitre appears on a seal of Archbisho p Thomas Becket (Father Thurston, The ?P allium, London, 1892, p. 17), The custom was, however, .already growing up of setting the horns over the front and back of the head instead of the sides (the mitre said to have belonged to St Thomas Becket, now at Westminster Cathedral, is of this type), 1 and with this the essential character of the mitre, as it persisted through the middle ages, was established.Advertisement
- 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).
- 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."Advertisement
- But 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.
- Cephalic disk enlarged anteriorly, forming an open tube posteriorly; shell external, thick, with p:ominent spire; no operculum.
- 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.Advertisement
- Y Y P > > much skill, elaborated from them the excellent work known as Nitzsch's Pterylographie, which was published at Halle in 1840, and translated into English, for the Ray Society in 1867.
- 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.
- No 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.Advertisement
- 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 ?
- A 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.
- This paper is principally based on the following general theorem, which is a remarkable extension of Pascal's hexagram: "If a polygon move so that each of its sides passes through a fixed point, and if all its summits except one describe curves of the degrees m, n, p, &c., respectively, then the free summit moves on a curve of the degree 2mnp. ..
- We put e for the eccentricity of the ellipse, represented P, by the ratio M CF: CA.
- 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.
- Representing 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.
- When Athens became the capital in 1833 the ancient name of The P 33 Peiraeus.
- Group V.: N, trivalent and pentavalent, but divalent in nitric oxide; P, As, Sb, Bi, trivalent and pentavalent, the last being possibly divalent in BiO and BiC1 2.
- 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.
- Mitscherlich, 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.
- By taking appropriate differences the following facts will be observed: (1) the replacement of potassium by rubidium occasions an increase in the equivalent volumes by about eight units, and of rubidium by caesium by about eleven units; (2) replacement in the same order is attended by a general increase in the three topic parameters, a greater increase being met with in the replacement of rubidium by caesium; (3) the parameters x and, p are about equally increased, while the increase in w is always the greatest.
- 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.
- In the Titanotheroidea the dentition may be expressed by the formula i: oru, ci, p T 4,, my.
- The dentition is i, c 1, p 4, m L total 42.
- Fully realizing, the difficult P Y g?
- If a brief definition of instinct, from the purely biological point of view be required, that given in the Dictionary of Philosophy and Psychology may be accepted: "An inherited reaction of the sensori-motor type, relatively complex 3'p 3' p and markedly adaptive in character, and common to a group of individuals."
- 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.
- To the east of the Yablonoi border-range p l ateau: lies the lower terrace of the high plateau, reaching 2000 to 2500 ft.
- 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.
- We can solve these, assuming them independent, for the - i ratios yl, y2,...yn-iï¿½ Now a21A11 +a22Al2 ï¿½ ï¿½ ï¿½ = 0 a31A11+a32Al2 +ï¿½ ï¿½ï¿½ +a3nAln = 0 an1Al1+an2Al2 +ï¿½ï¿½ï¿½+annAln =0, and therefore, by comparison with the given equations, x i = pA11, where p is an arbitrary factor which remains constant as i varies.
- 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.
- The law of reciprocity shows that p(s) = zti (m 1te2tmtL3t) t=1 st It 2t 3t viz.: a linear function of symmetric functions symbolized by the k specifications; and that () St =ti ts.
- The introduction of the quantity p converts the symmetric function 1 2 3 into (XiX2X3+...) -Hu Al (X 2 A 3 .-) +/l02(X1X3.ï¿½.) +/103(A1X2.ï¿½.) +....
- Denote by brackets () and [] symmetric functions of the quantities p and a respectively.
- P y q +...
- Writing D = gi d od p!
- P141 P242 ï¿½ï¿½ï¿½ -1 hp, - hpg = is converted into where dlo = d a P,q-1 - dapg vanish.
- 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.
- The existence of such forms seems to have been brought to Sylvester's notice by observation of the fact that the resultant of of and b must be a factor of the resultant of Xax+ 12 by and X'a +tA2 for a common factor of the first pair must be also a common factor so we obtain P: = of the second pair; so that the condition for the existence of such common factor must be the same in the two cases.
- 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.
- The General Term Of A Seminvariant Of Degree 0, 0 And Weight W Will Be A A A Appb°Ob°1B°2...B°4 _ 0 1 2 P 0 1 2 Q P Q P Q Where Ep S =0, Eas=0 And Esp, A Es,=W.
- 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 ï¿½, In 1 Z 2 1 Azp. 1 Azp 2....1 A2 P 2.1 Az P .
- Thus, For Two Linear Forms, P =Q = I, We Find 1 Z 2 1 Az.
- 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.
- It is usually prepared by the so-called "Reimer" reaction (Ber., 1876, 9,p. 1268), in which chloroform acts on phenol in the presence of a caustic alkali, C 6 H 5 OH+CHC1 3 +4KHO = 3KCI+3H20+KOï¿½C6H4ï¿½CHO, some para-oxybenaldehyde being formed at the same time.
- 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.).
- His authority, was absolute p 3'> too, > being tempered only by the shadowy right of the Magyar nation to meet in general assembly; and this authority he was careful not to compromise by any slavish imitation of that feudal polity by which in the West the royal power was becoming obscured.
- 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.
- Ferdinand was elected (Dec. 16) by a scratch assembly consisting of deputies from Croatia and the towns Ferdinand of Pressburg and Sopron; but he speedily improved °fAustr;a g P Y P elected.
- 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.
- The theory may be extended to the cases of p= i and p = o; so that a 3 means a.a.a.1, a 2 means a.a.i, a 1 means a.i, and a° means I (there being then none of the multipliers a).
- 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.
- The principles of arithmetical transformation follow from those stated in ï¿½ï¿½ 15-18 by replacing X, A, B, m, M, x, n, a and p by any expressions involving or not involving the unknown quantity or number and representing positive numbers or (in the case of X, A, B and M) positive quantities.
- 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.
- The application of the method to the calculation of (I +x) n, when n= p/q, q being a positive integer and p a positive or negative integer, involves, as in the case where n is a negative integer, the separate consideration of the form of the coefficients b 1, b 2, ...
- 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.
- If ABCD is a tetrahedron of reference, any point P in space is determined by an equation of the form (a+13+ - y+5) P = aA+sB +yC +SD: a, a, y, b are, in fact, equivalent to a set of homogeneous coordinates of P. For constructions in a fixed plane three points of reference are sufficient.
- Classes: Arachnida, Insecta (including Sub-Classes M y riapoda, Hexapoda), Crustacea (including Sub-Classes Entomostraca, Malacostraca), Epizoa (Epizootic Crustacea), Annellata (Chaeto p ods and Leeches), Cirripedia.
- 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 dS =27rxdx, we have for the whole effect 27r œ sin k(at - p)x dx, f P ' or, since xdx = pdp, k = 27r/A, - k fr' sin k(at - p)dp= [- cos k(at - p)]°° r.
- 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.