p

He'd better watch his P's and Q's.

76ft., are round or square, and for these sizes, and shapes, and of course for a flat surface, the relation P = .003 is fairly correct.

33Open-air conventicles were held in all parts of the provinces, and the fierce Calvinist preachers raised the religious excitement of their hearers to such aitch that it found vent in a furious outburst The lcono- P oasts.

23The Spaniards skilfully avoided a battle, and in Alva November the invaders were compelled to withdraw triumph- p ant.

23Groen van Prinsterer, Archives ou correspondance ine'dite de la maison d'0range, P serie (9 vols., 1841-1861); Poullet et Piot, Correspondance du cardinal Granvelle (12 vols., 1879-1899); J.

23then to the paying-out drum P, from it to the dynamometer D, and finally to the stern pulley, over which it passes into the sea.

11The whole system provides the means of giving sufficient back-pull to the cable to make it grip the drum P, round which it passes several times to prevent slipping.

11On the same shaft with P is fixed a brake-wheel furnished with a powerful brake B, by the proper manipulation of which the speed of paying out is regulated, the pull on the cable being at the same time observed by means of D.

11The shaft of P can be readily put in gear with a powerful engine for the purpose of hauling back the cable should it be found necessary to do so.

11The length paid out and the rate of paying out are obtained approximately from the number of turns made by the drum P and its rate of turning.

11The earliest successful form was " Bright's bell " sounder, which consisted of two bells of distinct tone or pitch, one of which was sounded when the current was sent in one [[International Code O]] --- 4 - 5 p-- - 6 R - 7 '...'

11joined between P and Q.

11The P Line Receiving Instrument R FIG.

11One very great advantage in this method is that the instrument used between P and Q may be of any ordinary form, i.e.

11Each J~ 1 1 / Stelein bundle has its own ~ investment of tissue P corresponding with external conjunctive, and now called peridesm.

11Apical cell, p. Wall marking limit between the plerome k, initial segment of root-cap. P and the pleriblem Pb.

11This baptism may not be conferred until the candidate has reached an age of discrimination and capacity to remember obligations, p y cere seven years being fixed as the earliest age, but it is generally deferred until manhood.

11P. P.)

14By a modification of the bridge method, applied with excellent results by Dr Muirhead to submarine work, condensers are substituted for a and b, one being also placed in the circuit between P and Q.

0028, and the levers are adjusted so that the left-hand one moves a, b, c and punches a row of holes across the paper (group i in the figure), the middle one moves b only and punches a centre hole (2 in the figure), while the right-hand one moves a, b, d, e and punches O p p Oa Oa' Ob Od 0?

00A and A' by means of two metal pins P, P'.

00The average total inductive value of these bridges to received signals is about 40 henrys, and the coil is so arranged that the arms contain three sections or blocks of winding each, two of which are joined up to strap connexions, and the a p :?; .?

00Elec. Eng., 2 7, p. 869.) In 1899 experiments were made atMenai Straits to put the lighthouse at the Skerries into communication with the coastguard station at Cemlyn.

00Elec. Eng., 27, p. 938.) It may be explained as follows: - Suppose a battery on shore to have one pole earthed and the other connected to an insulated submarine cable, the distant end of which was also earthed; if now a galvanometer is inserted anywhere in the cable, a current will be found flowing through the cable and returning by various paths through the sea.

00Elec. Eng., 27, p.

00(Id., 27, p. 852.) In addition to the systems of wireless or space telegraphy depending upon conduction through earth or water, and the in ductive system based upon the power of a magnetic Eelson.

00P: M FIG.

00"3 Schafer, Science Abstracts, 1901, 4, p.

00Ann., 1890, 40, p. 56) employed an arrangement as follows: Four fine platinum or iron wires were joined in lozenge shape, and two sets of these R and S were connected up with two resistances P and Q to form a bridge with a galvanometer G and battery B.

00- A, antenna; P S, jigger or oscillation transformer; C, condenser; 0, Fleming oscillation valve; B, working battery; T, telephone; R, rheostat; E, earth-plate.

001 p, G r, 2 s, X t and d, V u and o, 8 f, d s (i.e.

00The four telephones on a circuit are so wired that the relays 9-- P ..,, connect two of the bells between each wire and fl-- 0 7-..9 *"y earth, and further that one of each pair of bells responds to positive and the other to negative o-- pulsations.

00Inasmuch as the debenture stocks and preference shares would have to be redeemed in 1911 at premiums ranging from 3 to 5 per cent., the state would have to pay the company £253,000 in excess of the total of the outstanding securities in order to enable the ordinary shares to receive par, and in the council's view this payment would diminish the p robability of the Post Office being able to afford a substantial reduction in the telephone charges.

00p, Pistillate, s, staminate flowers; h, sterile flowers forming a circlet of stiff hairs closing the mouth of the chamber formed by the lower part of the spathe.

00- Stauridium productum, portion of the colony magnified; p, polyp; rh, hydrorhiza.

00B, C. multicornis, natural size; p, polyp; gon, gonophores; rh, hydrorhiza.

0012 p 9 6' 3.

00Maas in Results of In its arrangement the muscular tissue the "Albatross " Expedition, forms two s stems: the one composed Museum of Comparative Y P Zoology, Cambridge, Masse, of striated fibres arranged circularly, that U.S.A. is to say, concentrically round the central FIG.

00to enclose the club in a protective covering form- ?0 ing a cup or vesicle, at first open distally; finally the opening closes and J' P g, ,.

00in this sub-order have received Such are the " snake-like zoids " and as such are generally inter 4"0 ' 'Y p P After Allman, Gymnoblastic Hydroids, by permission of the council of the Ray Society.

00- p elagic floating Hydrozoa with great differentiation of parts, each performing a special function; generally regarded as colonies showing differentiation of individuals in correspondence with a physiological division of labour.

00p, Pneumatophore.

00(1750), p.

00BM- Within quite recent years, however, a special school q Y P has arisen with the main object of treating the processes of evolution quantitatively.

00The diocesan synod P Y Y system.

00The relations of their bishops, priests or other ministers and lay office-bearers inter se and to their lay folk depend upon contract; and these Y P P contracts will be enforced by the ordinary courts of law.

00equivalent reQaapa?coo-n) (now superseded by the term p roTEta " the fast"), are derived from the Sunday which was the fortieth day before Easter, as Quinquagesima and Sexagesima are the fiftieth and sixtieth, Quadragesima being until the 7th century the capul jejunii or first day of the fast.

00P, Part of spiral hydroid (tracheid) of Phanerogam (Flowering Plant).

00(p) Assimilating (palisade) cells.

00P

00cluding bush-forest in Africa and campos serrados in Brazil P

00The natural su p position that the earth Greek .

00The Arctic voyages of Barents were quickly followed by the establishment of p u a Dutch East India Company; and the Dutch, ousting the Portuguese, not only established factories on the mainland of India and in Japan, but acquired a preponderating influence throughout the Malay Archipelago.

00The 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%.

00The acid renders it available as a manure by converting the calcium phosphate, Ca 3 P 2 O 8, that it contains into the soluble monocalcium salt, CaH 4 P 2 O 8, or "superphosphate."

00p, Parietal.

00P tO u:..,.. ?

00It has several marked deficiencies compared with Australia, among which are the babblers (Timeliidae), weaver birds (Ploceidae), the Platycercinae among parrots, diurnal birds of p rey and the emeus.

00In the exile, but probably after 50o B.C., an important section of the Hexateuch, usually called the Priest's Code (P), was drawn up. At various times in the same century are to be placed the book of Job, the post-exilic parts of Isaiah, the books of Joel, Jonah, Malachi and the Song of Songs.

00P ?

00In the Geonic period there came into prominence the sect of the Karaites (Bene migra, " followers of the Scripture", the Protestants of Judaism, who rejected rabbinical authority, basing their doctrine and practice exclusively on The g P Y ICaraltes.

00He 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.

00From 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.

00Disulphuryl 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.

00p apal s.

00On 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.

00Thus 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.

00P.

00In addition to this there are certain writings by his son Isidorus H€pc irpoaOuous, liuxiis; EO iy17ruca on the prophet Parchor (HapXci p); 'HBcrca.

00(P. A.)

00Y 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.

00Each province of the empire, except the now disfranchised steppes of Central Asia, 7 returns a certainro ortion of members (fixed in each case by P P (Y law in such a way as to give a preponderance to the Russian element), in addition to those returned by certain of 2 M.

00As 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.

00Every industrial concern employing fifty hands or over elects one or more delegates to the electoral P ?

00': opal e ° °o T A R ple ' ag a ',ap iJ,wl Karkinit A C K r B L Scale, English Miles D S E A 32 Stavropol P O L A PI A N L A s E Derbent ° I?

00By 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.

00p principles: the separation of the judicial and administrative functions, the independence of the judges and courts, the publicity of trials and oral procedure, the equality of all classes before the law.

00Those 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 ?'

00p p These had been detected and pointed out by learned ecclesiastics of Kiev, where some of the ancient learning of Byzantium had been preserved, and Nikon determined to make the necessary corrections.

00obtained for the Roman Catholics certainrivile es with regard to the Holy Places in The Holy P g ?'

002 The law establishing individual peasant-proprietorship was p assed on December 21st.

00More recently legislation has beenassed to safeguard the lives and interests of Hours of P g Lab ur.

00P 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.

00The 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).

00Where, as at a double-line junction, one pair of rails crosses another pair, " diamond " crossings (p) are formed.

00Between 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.

00The 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.

00Let 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.

00The 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.

00Assuming that the mean pressure in the other cylinder is also p, the total work done per revolution is 4pla.

00Hence, 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.

00If p is the mean pressure at any speed the total tractive force which the engine is exerting is given by equation (25) above.

00per second, the total resistance R, which the engine can overcome at this s p eed, is by equation (10) R=(1190X550)/88=7.400 lb.

00For a stated value of the boiler pressure and the cut-off the mean pressure p is a function of the piston speed v.

00164.) Substituting this value of p in (27) I.H.P. _ (c av (29) 550 the form of which indicates that there is a certain piston speed for which the I.H.P. is a maximum.

00This 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.

00The 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.

00(Leben des Ministers Freiherrn vom Stein (6 vols., 1849-1855); also, in an abridged form, Aus Steins Leben (2 vols., 1856) ° Scale, 1:12,000,000 English Miles p 50 Boundaries of Departments & Provinces Capitals Departments & Prouinces Railways Tumbez Trujillo

00The 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.

00The view of Professor Dillmann, who placed P before D in the regal period (though he admitted exilic and postexilic additions in Exod., Levit.

00On the other hand, according to P (Ex.

00It should be noted that in P (Code of Holiness) Lev.

00p: 564 (ed.

00To 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.

00To this age we may ascribe the literature of the Priestly writers (symbolized by P), which differs markedly from the other sources.

00Wolf, Menasseh ben Israel's Mission to Cromwell, p. lxxv.).

00It is clear that the rulers, as so P p commonly in ancient states, fulfilled priestly as well as royal functions.

00About 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.

00And as in Hebrew, the six letters b g d k p t are aspirated when immediately preceded by any vowel sound.

00Such 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.

00They have continued to be worn, however, by the bishops of the Scandinavian Lutheran Y P Churches.

00P - P.)

00(P. VI.)

00p1, pl, The two lateral divisions p, p,1, The three inner, of the inner perianth.

00p-- st--- FIG.

00p, Petals.

00The 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).

00Brunelli in L'Arte, 1907, p. 47.) The crypt contains three ancient sarcophagi.

00Among the Terebelloidea there is a remarkable differentiation of the ne p hridia into two series.

00b.z, Budding zone; p, anterior region of the parent worm; 1-5, buds.

00Let AB be the major axis of the orbit, B the pericentre, F the focus or centre of motion, P the position of the body.

00Drop the perpendicular RPQ through P, the position of the planet, upon the major axis.

00- (Residence at the Court of London, p. 286.) Bentham's love of flowers and music, of green foliage and shaded walks, comes clearly out in this pleasant picture of his home life and social surroundings.

00One of the most p im ortant ways of keeping insect pests in check is by " spraying " or " washing."

00But in his character as phenomena must be examined or what may be neglected p y g in economic inquiry.

00burr.- p T 9 pl.y ped.g: reversal of the cleavage planes in sinistral as compared with dextral forms. The facts, however, strongly suggest that the original cause of the torsion was the weight of the exogastric shell and visceral hump, which in an animal creeping on its ventral surface necessarily fell over to one side.

00k, 1, p, J affords a means of test ing the conclusion that we have in Lankester's 4 capito-pedal bodies the rudimentary ctenidia.

00p, Pedal nerve.

00p, Snout.

00p, Penis.

00P, Pedal ganglion with otocyst attached.

00P sp, Supra - intestinal visceral ganglion on the course of the right visceral cord.

00P, A, FIG.

00p, Liver.

00- P l e u r o c e r i d a e.

00k, Opening of the albuphrodite duct, which very soon becomes miniparous gland into P Y the hermaphrodite entwined in the spire of a gland - the duct.

00greatest differences as to the amount p of food-material by which the egg-shell is encumbered.

00Cephalic disk enlarged anteriorly, forming an open tube posteriorly; shell external, thick, with p:ominent spire; no operculum.

00o and p, The liver.

00In the district, and es p ecially at Salobrena, 3 m.

00- Thorax of Saw-Fly (P I, Dorsal view.

00p, Coxa of hind leg.

00p, Pericardial septum.

00p YP, gg FIG.

00The 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.

00p. xcvii.).

00Among contem p orary writers in a more popular style are John Burroughs; Herbert K.

00Y 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.

00This 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.

00No 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.

00The dates of the various parts of the existing ducal palace are indicated on the plan; the rebuilding was carried on in the following order, P, Q, R, S, T, U, V.

00The pedal equation is r 3 =a 2 p, which shows FIG.

00According 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.

00An original medial p preceding the chief accent of the word also appears as b in English and the other members of the same group. It is not certain that any English word is descended from an original word beginning with b, though it has been suggested that peg is of the same origin as the Latin baculum and the Greek (6 KTpov.

00(P. G1.)

00y P P to assume heavy risks.

00The work of the committee is by no means simple, as frequently very few transactions take place p ?

00p in the kinds of cotton of which quotations are given.

00This question can readily be answered as regards the past forty years or so, for which material g p y y ?

00P, Proboscis.

00' °° p Q; i r ' Afayette A(l.

00P f'

00y, P J ° c.

00> Ft.Drum O,p t ?S 1 1 ?

00p ff"Nocatee ti Braden n atee ?i ?

00A 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."

00Its consonants are k, g, ng, ch, j, n, t, d, n, p, b, m, y, r, l, w, s, h.

00This 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. ..

00P. A.)

00We put e for the eccentricity of the ellipse, represented P, by the ratio M CF: CA.

00Let P, P' be two consecutive positions of the radius vector.

00Since the area of the triangle FPP' is one half the product of FP into the perpendicular p from P on FP', it follows that if these perpendiculars were equal all round the orbit, the areas described during the infinitesimal time would be smallest at the pericentre and continually increase during the passage of the body to B.

00It follows that p must be greatest at pericentre, where its distance from F is least.

00By 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.

00Representing 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.

00The p y ?

00g p Atticus in memory of his wife Regilla, is comparatively well preserved; it was excavated in 1848 and in 1857-1858.

00When Athens became the capital in 1833 the ancient name of The P 33 Peiraeus.

00Bibliography.-C. Imbault-Huart, L' p le Formose, histoire et description (Paris, 1893), 40; J.

00in which to keep his 6, 6, North and south vaulted transepts p (the dotted lines show the curve clothes and books; these of the vault).

00g p p from the earliest times.

00Group 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.

001.2 or 1.6, named ortho- (o), 1.3 or 1.5, named meta- (m), and 1.4, named para- compounds (p).

00Di-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.

001 7, p. 1 347), who has also suggested the use of manganese carbonate instead of magnesite, since the change of colour enables one to follow the decomposi 411=ThEIZ P; FIG.

00Eliminating 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.

00If, 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.

00These relations may be readily tested, for the ratio C p /C„ is capable of easy experimental determination.

00Let double bonds be present, in number p, and let the energy due to such a bond be Y.

00' C 6 Hcooh C?H 4`C'p C s H COOH Polymorphism.

00Mitscherlich, 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.

00By 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.

00S, Se; Te (in tellurides); Cr, Mn, Te (in the acids H 2 RO 4); As, Sb (in the glances MR2) As, Sb, Bi; Te (as an element); P, Vd (in salts); N, P (in organic bases).

00In 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.

00(P. Gx.)

00Lumby (Cambridge, 1883), supplemented a little„by Edward Hall (Chronicle, p p. 3 6 3-3 6 4).

00A distinguished philosopher or man of letters would find them bidding f o r his presence, and most of the great names are p ?

00Ducarla, 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.

00O IXC a P orrelln, nafurr d Yae?a?

00_? ?w?lan F?,o i o a p oba - ?

00Of SchOner we know that he produced four globes, three printed from segments (1515, 1523, 1 533), and p SCF12MER.S FIG.

00This map was repeatedly revised, Antgria- g P P Y ?

00Although 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.

00woman to offer vocal prayer, to read the Scriptures, p or to utter such exhortation or teaching as may seem to be called for.

00` Hv6 X aQOV µ7 7 EA p s.

00This identification can only be p urchased at the cost of a complete renunciation of the Avestan genealogy.

0040p i i), pasture, (b p(3ecv, to feed, Sans.

0010), protected ° - - from irreverence by lattice work (tra p - ` j '.?- ' `- sennae) of marble.

004 p .`3 D ° a d a D ?aD?paD paoDL,?

00p ©?ii 'o ?e0 ?? ?

00In the Titanotheroidea the dentition may be expressed by the formula i: oru, ci, p T 4,, my.

00The dentition is i, c 1, p 4, m L total 42.

00Fully realizing, the difficult P Y g?

00If 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."

00Originally - P Y g Y residents at Santiago de Cuba, the captains-general resided after 1589 at Havana.

00Piron, L' p le de Cuba (Paris, 1876).

00P For sulphonic acids containing nitrogen see Ammonia.

00Berthelot, Comptes rendus (1907), 1 44, P z69

00(P. Vi.)

00p I .,' a I ??

00p n, 5 1 ?

00'P V y ndina st .

00P Mctso^o Gr/ 4 ?

00bn a p.

00When 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.

00P 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.

00He 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.

00Y P P importance.

00p expenditure, facilitated by the enthusiasm created in Europe by Turkey's admission to the ranks of the powers which loosened for her the purse-strings of the foreign investor.

00p. 197.) From 1890 Crete was frequently the scene of disturbance., the Christian communities in other parts of Turkey began to.

00Their 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.

00P. Pick; but in the present state of our knowledge, however, p g > the older classification of E.

00In 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.

00Chemie, 1889, p. 96; and 1892, 9, p.

00The 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.

00On 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.

00Yet later the co p e seems to have been authoritatively proscribed with the rest.

00Tubes 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.

00To the east of the Yablonoi border-range p l ateau: lies the lower terrace of the high plateau, reaching 2000 to 2500 ft.

00P, Peduncle.

00On 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.

00b., we obtain by comparison with the middle series the symbolical representation of all symmetric functions in brackets () appertaining to the quantities p i, P2, P3,ï¿½ï¿½ï¿½ To obtain particular theorems the quantities a l, a 2, a 3, ...a, n are auxiliaries which are at our entire disposal.

00p, g 5

00Since the determinant having two identical rows, and an3 an3 ï¿½ï¿½ï¿½ ann vanishes identically; we have by development according to the elements of the first row a21Au+a22Al2 +a23A13+ï¿½ï¿½ï¿½ +a2nAin =0; and, in general, since a11A11+a12A12 +ai 3A13+ï¿½ï¿½ ï¿½ +ainAin = A, if we suppose the P h and k th rows identical a A +ak2 A 12 +ak3A13+ï¿½ï¿½ï¿½ +aknAin =0 (k > i) .and proceeding by columns instead of rows, a li A lk +a21A2k + a 31A3k+ï¿½ï¿½ï¿½+aniAnk = 0 (k .>

00so that A breaks u p into a sum of determinants, and we also obtain a theorem for the addition of determinants which have rows in common.

00We 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.

00Resultant 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.

00For if u, v, w be the polynomials of orders m, n, p respectively, the Jacobian is (u 1 v 2 w3), and by Euler's theorem of homogeneous functions xu i +yu 2 +zu 3 = mu xv1 +yv2 +zv3 = /IV xw 1+y w 2+ zw 3 = pw; denoting now the reciprocal determinant by (U 1 V2 W3) we obtain Jx =muUi+nvVi+pwWi; Jy=ï¿½.., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J.

00Further, if m '=' p, we obtain by differentiation 7+x =m (u;1-2xl.

00p 1 p 2 ...p n), it is necessary to bring under view other functions associated with the same series of numbers: such, for example, as P P3 P2 P4 Pn -2 /, /, /, F i a i 1 a 2 Fi a 1 a 2 ...

00A partition, (pipipip2p2p3) = can be separated in the manner (p 1 p 2) (PIP2) (p1P3) = (1)12,2) 2 (plp3), and we may take the general form of a partition to be (pi i p2 2 p3 3 ...) and that of a separa tion (J 1) 1 1(J 2) 5 2(J 3) 1 3...

00(P i +Y2+t' +...-1)!

00ï¿½ ï¿½ P1 v2 v3...) ï¿½ For, writing as before, Xm 'Xm 2 Xm '= zzo(SQls:2s73...) xi'x12x13..., 1 2 3" 1231 2 3 = EPxi l x A2 x A3, P is a linear function of separations of(/ 1 / 2 A2 / 4 3 3 ...) of specification (m"`1mï¿½2m"`3...), and if X; 1 X 3 2X8 3 ' ..

00The 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.

00A table may be formed expressing the k expressions Pa l), P(2),...P(1) as linear functions of the k expressions (m"`'smï¿½2smï¿½3s...), s =1, 2, ...k, and the numbers BSc occurring therein is 2s 3s possess row and column symmetry.

00P(s) _ /ll8!/12s!/23s!...

00appertaining to the function (li'l32l3...), each separation having a specification m" ` ' 8 m ï¿½2s m ï¿½38 multiply b P (is 2s 3s .ï¿½ï¿½), P Y by ls!

00so that f (al, a 3, a3,.ï¿½.an) =f, a rational integral function of the elementary functions, is converted into f(a1 +12, a2+ p a1,...

00Write also s l d1= D, so that f(a i a2+ p al, ...an+Ilan-1) =f +FLDif +F4 2 D2f + t i 3 D 3 f -}-....

00The 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.) +....

00p operators D upon a monomial symmetric function is clear.

00+AsDs ...) p Di p2 D2+...

00+an we may write in general D s f = ZD(p l p 2 p 3 ï¿½ï¿½) the summation being for every partition (piP2p3...) of s, and D(p iP2 p 3 ...)f being =2 (Dpifi)(DP2f2) (DL'h3f3)f4...f,n.

00We have the theorem (I I, v; m, n) (/l l, v l; ml, n i) - (Il l, P 1; m l, n ') (/l, v; m, n) = (11, vl; ml, ni); where 1 /l1= (ml +m-1) ml (/l +nlv) - u-2 Cu '+nvl) 1 1 m-1 1 m1-1 vl =(n -n)vv-E ml / lY- m /lv, m i =7111+m-I, n1=nl+n, and we conclude that qua " alternation" the operators of the system form a " group."

00p, taking n equal to 00, we may write 1 +aix +a2x2 +...

00_ (1 + p ix) (1 + P2x) ...

00Denote by brackets () and [] symmetric functions of the quantities p and a respectively.

00Then 1111 + a i[ 1 ]+ a i [12 1+a2[ 2 ]+ a 7 [13] +ala2[ 21 ]+a3[3]+-ï¿½ + a p1 a p2 a P 3 ï¿½ï¿½ .ap rn[Y1 p 2t' 3 ...

00+amaa.)sm, (ll, v; m P -"O a an + (l l + v) (ll +2 v) (m (11 +3v) +...], m - 2 2 !2 a 0 a 1 aan +2 !

00ï¿½, and (alai +(72a2+a3a3+ï¿½ï¿½ ï¿½) P = (Pith +P2t2 +P3f 3 3+ ï¿½ ï¿½ ï¿½) P ï¿½ Instead of the above symbols we may use equivalent differential operators.

00P y q +...

00/,) (-)P+4-laP4 (p i+ g l - 1) !

00Further writing 1 +hlox+holy+...+ hpgx P y {-...

00Writing D = gi d od p!

001 exp(Adlo + vdol) = (1+/oD10+ v Doi +..ï¿½+ VQ +.ï¿½.)f; now, since the introduction of the new quantities 1.1., v results in the addition to the function (plglp2g2p3g3...) of the new terms A PI Pg1 (p 2q2 p 3g3ï¿½ï¿½ï¿½) +/ AP2Pg2 (p 1 g 1P343 ...)+/ Z3vg3 (p l g i p 2 g 2 ...)+ ï¿½, we find DP141(plqip2q2p3q3ï¿½ï¿½ï¿½) = (p 2 q 2 p 3 q 3ï¿½ï¿½ï¿½), and thence D P141 D P242 D P343 ï¿½ï¿½.

00(p g p, g p, g3 ï¿½ï¿½ï¿½) = I; while D rs f =o unless the part rs is involved in f.

00P141 P242 ï¿½ï¿½ï¿½ -1 hp, - hpg = is converted into where dlo = d a P,q-1 - dapg vanish.

00In the present particular case putting m 10 = 1 2, mot= v and m P4 =o otherwise M10t+M01n+...+Mpot P n 4 +...

00=log (1 +ï¿½t+vn) M P4 = (_)p+4 -1(p+g 1)!ï¿½p p 4; p!g!

00+1,q p l !gll p2!g2!

00Recalling 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.

00p. 493) that the operation 1 1 d P?

00We may remark the particular result (-) p + p q!

00Since 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.

00=d P 4 lodp+1,4 -holdp,4+1+...+(-)r+shrsdp+r,4+s+..

00)- (p+4-1 (p - - q -1)!dpq+ ?l -) 1)!D'1 DT2 p!g!

00the i t " power of that appertaining to a x and b x multiplied by the j t " power of that appertaining to a x and c x multiplied by &c. If any two of the linear forms, say p x, qx, be supposed identical, any symbolic expression involving the factor (pq) is zero.

00possess the invariant property, and we may write (AB) i (AC)'(BC) k ...A P E B C...

00The 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.

00The Hessian 0 =A 2 is such that (f, 2 and if f is expressible in the form X(p x) 3 +,i(g x) 3, that is as the sum of two perfect cubes,.

00we find that Di must be equal to p x g x for then t x (p x) 3 +, u (g x) 3, Hence, if px, qx be the linear factors of the Hessian 64, the cubic can be put into the form A(p x) 3 +ï¿½(g x) 3 and immediately solved.

00For it is easy to establish] the formula (yx) 2 0 4 = 2f.4-2(f y 1) 2 connecting the Hessian with the quartic and its first and second polars; now a, a root of f, is also a root of Ox, and con se uentl the first polar 1 of of q y p f?

00Remark.-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.

00v., established the important result that in the case of a form in n variables, the concomitants of the form, or of a system of such forms, involve in the aggregate n-1 classes of aa =5135 4 +4B8 3 p) =0, =5(135 4 - 4A 2 p 4) =0, P yield by elimination of S and p the discriminant D =64B-A2.

00When 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.

00Solving 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.

00The 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.

001111 The Number Of Such Terms Is The Number Of Partitions Of W Into 0 0 Parts, The Part Magnitudes, In The Two Portions, Being Limited Not To Exceed P And Q Respectively.

00Denote This Number By (W; 0, P; 0'.

00The Number Of Linearly Independent Seminvariants Of The Given Type Will Then Be Denoted By (W; 0, P; 0', Q) (W; 0, P; 0', Q); And Will Be Given By The Coefficient Of A E B E 'Z W In L Z 1 A.

001 Ze'; Which Preserves Its Expression When 0 And P And 0 And Q Are Separately Or Simultaneously Interchanged.

00Taking 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 .

00Thus, For Two Linear Forms, P =Q = I, We Find 1 Z 2 1 Az.

00If Hebrew, it might be derived from the root p rr (to embrace) as an intensive term of affection.

00p, a thickened line on the walls forming the placenta; c, calyx; d, ovary; s, hooded stigma terminating the short style.

00p, placenta; o, ovules; s, suture, or median line of carpel.

00b, pair of bracteoles below the flower; s, sepals; p, petals; st, stamens; o, ovary.

00No 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.

00For the resultant force at P, F=-VF r 2.

00Since p can never be infinite, complete shielding is not possible.

00If 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.

00If 2l is the distance between the poles m and - in, d the distance from either pole to a point P on the line AB (fig.

005), we have for the resultant force at P R = - 2cosO X m/d' = - elm/d 3 = - M/d3.

00When P is the neutral point, H is equal and opposite to R; therefore M = Hd 3, or the moment is numerically equal to the cube of the distance from the neutral point to a pole, multiplied by the M.

00The 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.

00The 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.

00d3H P = d3H And I = v tan B.

00Denoting 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).

00The 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.

00The 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.

00The 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.

00P. Harrison.?

00The 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.

00(After Lankester, toe cit.) Magnified MOO p GC. D R.

00p 6 (90e00 b ", o o.

00o o 'e ' e c m_o 'o - _ - o i: (t __ '®moo?p ®04 ??a:a??

00p. 152.) in regard to the ovary, and by Benham (14) in regard to the testis.

00p O ??o?, ?_ 0 c?

00p, Pecten.

00p, Telsonic carapace.

00p, The tergal stigmata of the tergo-sternal muscles.

00The sperm is removed by the male from the genital aperture into a special receptacle on the terminal segment P FIG.

00Under side of the uplifted genital or first opisthosomatic somite of the female; g, genital aperture; p, pitted plate, probably a gland for the secretion of adhesive material for the eggs; 1, the edges of the lamellae of the lung-books of the first pair.

00p. 19.)

00It 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.

00r //,p, r ' an ho ?

00P .Antbpio S.Christovao sranaia imbo inhas antos ?

009 eneZd He .aop elo P-or o Alegre VP P'S Continuatio Southward ul (Same Scale) G 46° na, Sao own Para Blumenau 4d

00Among 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.

00The 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.

00According 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.).

00His 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.

00During 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.

00P g g the emperor Maximilian and Sigismund of Poland, might be dispensed with.

00Ferdinand 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.

00whole of Hungary except Syrmia and the territory g Y p Y Y the peace of Passarowitz (July 21, 1718), by which the Temeskaz was also freed from the Turks, and Servia, Northern Bosnia and Little Walachia, all of them ancient conquests of Hungary, were Once more incorporated with the territories of the crown of St Stephen.

00In 1823, when the reactionary powers were meditating joint action to suppress the - revolution in Spain, the government without consultin P ?

0026-2 Y P (26-27), was forced to retreat.

00On the 17th of February 1867 a responsible inde pendent ministry was formed under Count Gyula 'p y y ' of 1867.

00" F r ee Principle " party, left p) p y?

00P 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.

00Y, 9 P ?, PP ment of a Coalition cabinet 2 under Dr Sandor Wekerle was announced, the world was taken completely by surprise.

00playersg p heroes.

00Thus, 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.

00(b) If A=x times M, then x (iii.) (a) If n= P then x = ?i n.

00(iii.) (a) If p = log i n, then n = (b) If a = n, then n =a'.

00For 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.

00(a) If a = d, then each = p b qd.

00The 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).

00In n = a P, a is the root or base, p is the index or logarithm, and n is the power or antilogarithm.

00But a P is sometimes incorrectly described as " a to the power p "; the power being thus confused with the index or logarithm.

00It should be observed that, by analogy with the definition of a fraction, a P l q mean (al/q)P, not (aP)llq.

00(i.) To divide P by Q, we cut off from P successive portions each equal to Q, till we have a piece R left which is less than Q.

00Thus P = kQ+R, where k is an integer.

00Having 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.

00(iv.) If P and Q can be expressed in the forms pL and qL, where p and q are integers, R will be equal to (p-kq)L, which is both less than pL and less than qL.

00Hence u is the greatest common measure of p and q.

00In 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.

00is then an expression of the form a p b e c r d s, where p, q, r, s have the greatest possible values consistent with the condition that each of the given expressions shall be divisible by a p b e c r d s .

00is of the form a p b e c r d s, where p, q, r, s have the least possible values consistent with the condition that a P PC'd s shall be divisible by each of the given expressions.

00The 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.

00Thus from the equation P+Q - R+S=T and the identity P+Q - R+S= P - R +S+Q we have the equation P - R +S+Q=T, which is the same statement as P - R +S=T - Q.

00Thus from P+Q - R+S=T we deduce P+(Q - R+S)=P+(T - P).

00Thus, 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.

00The standard form is usually taken to be ax2+bx+c =0, from which we find, by transformation, (2 ax+b) 2 =b 2 - 4ac, 4 (}b 2 -4ac} -b and thence x = 2a (ii.) In an equation of the form Q=V, the expressions P, Q, U, V are usually numerical.

00It should be observed that, for r=o, (4) is replaced by n (o) _ (n - I)(o) (9), and similarly, for the final terms, we should note that P0)= 0 if 4> p (io).

00This 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.

00(b) Let us assume that the product of every set of p consecutive integers is divisible by p!, and let us try to prove that the product of every set of p+ I consecutive integers is divisible by (p+i)!.

00Then the assumption is that, whatever positive integral value n may have, n [P] is divisible by p!.

00But, by hypothesis, n 1P1 is divisible by p!.

00Therefore n 1P+11 -(n-I) (P+11 is divisible by p!.

00Therefore, if (n-i) 1P+11 is divisible by (p+I)!, 'n 1P+11 is divisible b y (p +i) !.

00(c) Thus, if the theorem of ï¿½ 41 (v.) is true for r= p, it is true for r= p+1.

00r!) = (n + r)(r) = (n+r)(n) (17)ï¿½ (iv.) By means of (17) the relations between the binomial coefficients in the form p (4) may be replaced by others with the The most important relations are n[r] = n[r-i]+(n - I)(r) (r8); O[r] = 0 (19); n[r]-(n-s)[r] =n[r-i]+(n- I) [r-1]+...+ (n-s+I)[r-1] (20); n[r] =n[r-1]+(n-I)[r-1]+...+I[r-1] (21).

00The number (n P r) of permutations of r individuals out of a stock of n, all being distinguishable, is n`r).

00If, out of every N cases, where N may be a very large number, a is A in pN cases and not-A in (I - p) N cases, where p is a fraction such that pN is an integer, then p is the probability or frequency of occurrence of A.

00