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aa

The southeastern corner of the province is traversed by the Westerwolde Aa, which discharges into the Dollart.

84+na n _ i aa n = 0.

53"Join AA," I answered.

45HELMOND, a town in the province of North Brabant, Holland, on the small river Aa, and on the canal (Zuid-Willems Vaart) between 'sHertogenbosch and Maastricht, 241 m.

33= 0; and such functions satisfy the differential equation aoaa i +2a0a 2 +3a 2 aa 3 +...

33the two quadratic forms f, 4); the two discriminants (f, f')2,(0,4')2, and the first and second transvectants of f upon 4, (f,, >) 1 and (f, 402, which may be written (aa)a x a x and (aa) 2 .

33By March 18 the Bolsheviks were thrown back over the Aa river.

33HELMOND, a town in the province of North Brabant, Holland, on the small river Aa, and on the canal (Zuid-Willems Vaart) between 'sHertogenbosch and Maastricht, 241 m.

33= wj, aa 1 aa 2 a a 3 the complete system of equations satisfied by an invariant.

34+a"aa"-1 have been much studied by Sylvester, Hammond, Hilbert and Elliott (Elliott, Algebra of Quantics, ch.

34+a"aa"-1 have been much studied by Sylvester, Hammond, Hilbert and Elliott (Elliott, Algebra of Quantics, ch.

34Aa 18 18

35A 1, A2 ï¿½ Ai, A 1 A 2, A2 and then Ao = al Ai+2a1a2AIA2+a2 A2 - (a1A1+a2A2) 2 = a?, A l = (a 1 A 1 +a2A2) (alï¿½l +a2ï¿½2) = aAaï¿½, A 2 = (alï¿½l +a2/-12) 2 = aM; so that A = aa l +2a A a u 152+aM5 2 = (aA6+a,e2)2; whence A1, A 2 become a A, a m, respectively and ?(S) = (a21+a,E2) 2; The practical result of the transformation is to change the umbrae a l, a 2 into the umbrae a s = a1A1 +a2A2, a ï¿½ = a1/ï¿½1 + a21=2 respectively.

35Now D A xA k = (n - k) A k; Aï¿½ A k = k A?1; D ï¿½A A k = (n - k) A k+1;D mï¿½ A k = kA k; (n - k)A ka - w Ak - 1 aA k = O; a _ J (n - k) A k +l A k = O; kA k Ak = wJ; equations which are valid when X 1, X 2, ï¿½ 1, ï¿½2 have arbitrary values, and therefore when the values are such that J =j, A k =akï¿½ Hence °a-do +(n -1)71 (a2aa-+...

23l aa k -x 2 d d- = 0; Z(nk)ak+l adk - x ldd2=0; or in the form d d 52-x 2(7 =0, O - x1ax2 = 0; where 0 = ao d a l + 2a 1 -?...+na,,_id an, 0 = nal dao -?

23By similarly transforming the binary n ic form ay we find Ao = (aI A 1 +a2 A2) n = aAn A l = (alAi - I -a 2 A 2) n1 (a1ï¿½1 +a2m2) = aa a ï¿½ - A i n-1 A2, n-k k n-k k n-k k A = (al l+a2A2) (alï¿½1+a2ï¿½2) = a A ï¿½ =A 1 A2, so that the umbrae A1, A 2 are a A, a ï¿½ respectively.

24- (aa) and then expresses the coefficients, on the right, in terms of the fundamental invariants.

24then of course (AB) = (ab) the fundamental fact which appertains to the theory of the general linear substitution; now here we have additional and equally fundamental facts; for since A i = Xa i +,ia2, A2= - ï¿½ay + X a2, AA =A?-}-A2= (X2 +M 2)(a i+ a z) =aa; A B =AjBi+A2B2= (X2 +, U2)(albi+a2b2) =ab; (XA) = X i A2 - X2 Ai = (Ax i + /-Lx2) (- /-jai + Xa2) - (- / J.x i '+' Axe) (X a i +%Ga^2) = (X2 +, u 2) (x a - = showing that, in the present theory, a a, a b, and (xa) possess the invariant property.

24(b) In the same way, for the expansion of (Aa)", let o, denote uo-u 1 +.

12If a, a be the linear forms, above defined, he raises the identity ax(0) =ax(aJ3) - (3x(aa) to the fifth power (and in general to the power n) obtaining (aa) 5 f = (a13) 5 az - 5 (a0) 4 (aa) ax?3 -F...

13Van der Aa, Biographisch woordenboek der Nederlanden (22 vols., Haarlem, 1852-1878).

14The minor Aik is aa, and is itself a determinant of order n-t.

14In some old roofs the rafters are connected without any intervening ridge-plate, with the result that after Sectional elevation on AA.

00The five fixed webs are attached to the table which is secured to the bottom of the box by the screws p. The three movable webs are attached to the projections XX on the frame aa.

00C AA.

00of Sa aa ?

00Such an expression as a l b 2 -a 2 b i, which is aa 2 ab 2 aa x 2 2 ax1' is usually written (ab) for brevity; in the same notation the determinant, whose rows are a l, a 2, a3; b2, b 2, b 3; c 1, c 2, c 3 respectively, is written (abc) and so on.

00To apply the method to the calculation of N n, it is necessary that we should be able to express N in the form A+a or Aa, where a is small in comparison with A, A" is easy to calculate and a/A is convenient as a multiplier.

00Each symbol a is associated with its supplement a which satisfies the equivalences a+a = i, aa = o, the latter of which means that a and a have no region in common.

00A point A in space may be associated with a (real, positive, or negative) numerical quantity a, called its weight, and denoted by the symbol aA.

00The sum of two weighted points aA, (3B is, by definition, the point (a+13) G, where G divides AB so that AG: GB =0: a.

00X is, in general, a determinate point, the barycentre of aA, 3B, &c. (or of A, B, &c. for the weights a, 0, &c.).

00+X) happens to be zero, X lies at infinity in a determinate direction; unless - aA is the barycentre of (3B, yC, ...

00The brightness (corresponding to B) midway between the two central points AA' is 8106 of the brightness at the central points themselves.

00If A' be the area over which the intensity must be Io in order to give the actual total intensity in accordance with A'102 =ff + 12d4dn, the relation between A and A' is AA' = X 2 f 2 .

00The position of Q corresponding to a given value of V, that is, to a band of given order, is by (19) BQ= aa b AD=V?

00HC:N ›NH N:CH ›NH HC:CH" Osotriazole (aa').

00If at any points of a vortex line the cross-section ABC, A'B'C' is drawn of the vortex filament, joined by the vortex line AA', then, since the flow in AA' is taken in opposite directions in the complete circuit ABC AA'B'C' A'A, the resultant flow in AA' cancels, and the circulation in ABC, A'B'C' is the same; this is expressed by saying that at all points of a vortex filament wa is constant where a is the cross-section of the filament and w the resultant spin (W.

00rf > a> b> o> a'> -Do; and then so that PT =c/Zir, and the curve AP is the tractrix; and the coefficient of contraction, or breadth of the jet breadth of the orifice - +i' A change of S2 and 0 into nS2 and nO will give the solution for two walls converging symmetrically to the orifice AA 1 at an angle zr/n.

00u -b' Along a jet surface, q=Q, and ch S2= cos 0 =cos a-i sin2a(a-a')/(u-b), (5) if 0 =-a at the source x of the jet xB, where u = co; and supposing 0=0,13 at the end of the streams where u =j, j', u-b i sin 2 a u - j cos 0-cos /3 i a -a cos a sin a -cos 0' aa' - 2 (cos a -cos (3) (cos a-cos 0)' u-j' 1 2 cos 0-cos, (6) a -a' - 2 S i n a (cos a -cos (3') (cos a -cos B)' and 4' being constant along a stream line d4 - dw ds _d8 d4 _ dw du du du' d- -dud0' 7rQ ds_ it ds (cos a-cos /3) (cos a -cos (3') sin 0 m+m' dB c d0 - (cos a-cos B) (cos 0-cos /3) (cos 0 -cos /3')' _ sin 0 cos a-cos 13 sin 0 - cos a-cos B + cos 0-cos (3' cos 0-cos 13 cos a -cos $ sin 6 cos (3-cos /3' cos 0-cos 0" giving the intrinsic equation of the surface of a jet, with proper attention to the sign.

00______ Pilgrim and Trade Routes Swamps EH B Aa Mediterranean Muwdla sko / - /2nd.

00The harbour is connected by canals with the river Aa and the navigable waterways of the department.

00and 655,800 inhabitants) (5) Parts of the former province of Courland between the old German frontier and the Holy Aa river, as also part of the district of Illuxt.

00Cramer, light has been thrown on the period by the researches of de Hoop Scheffer; see Van der Aa, Biographisch woordenboek der Nederlanden (1869) R.

00The classification of the different kinds of coal may be considered from various points of view, such as their chemical composition, their behaviour when subjected to heat aa s sifica= or when burnt, and their geological position and iron.

00(iii) Take any straight line intersecting or not intersecting the figure, and draw perpendiculars Aa, Bb, Cc, Dd,.

00Its surface is diversified by several plateaus, those of Haanhof and of the Livonian Aa having an average elevation of 400 to 700 ft., while several summits reach Boo to loon ft.

00The edges of the plateaus are gapped by deep valleys; the hilly tract between the Dvina and its tributary the Livonian Aa has received, from its picturesque narrow valleys, thick forests and numerous lakes, the name of "Livonian Switzerland."

00along its frontier, the Pernau, Salis, Livonian Aa and Embach are navigable.

00Live-stock and dairy products are important factors in the T C t o L %i, aa.

00on in time dt to A'B'C', where AA' = Udt, the displacement of the particle originally at M must change from PM to P'M or by PP'.

00Now taking equation (72), and replacing tan B, as a variable final tangent of an angle, by tan i or dyldx, (75) tan 4) - dam= C sec n [I(U) - I(u)], and integrating with respect to x over the arc considered, (76) x tan 4, - y = C sec n (U) - f :I(u)dx] 0 But f (u)dx= f 1(u) du = C cos n f x I (u) u du g f() =C cos n [A(U) - A(u)] in Siacci's notation; so that the altitude-function A must be calculated by summation from the finite difference AA, where (78) AA = I (u) 9 = I (u) or else by an integration when it is legitimate to assume that f(v) =v m lk in an interval of velocity in which m may be supposed constant.

005, Aa and H).

00i (5), one of the labial tentacles n is also thrown back to show the mouth w, and the two left gill-plates are reflected to show the gill-plates of the right side (rr, rq) projecting behind the foot, the inner or median plate of each side being united by concrescence to its fellow of the opposite side along a continuous line (aa).

00374: "AA represents a small scale.

004, when it ought properly to sink almost to the top of the stem, the surface tension of the water around the circumference of the circle of contact, AA', providing the additional support required.

00The three chief rivers of the province are the Elbe in the north-east, where it mainly forms the boundary and receives the navigable tributaries Jeetze, Ilmenau, Seve, Este, Luhe, Schwinge and Medem; the Weser in the centre, with its important tributary the Aller (navigable from Celle downwards); and in the west the Ems, with its tributaries the Aa and the Leda.

00van der Aa, Biog.

00The surface of the upper slopes of Mauna Loa is almost wholly of two widely different kinds of barren lava-flows, called by the Hawaiians the pahoehoe and the aa.

00The aa is lava broken into fragments having sharp and jagged edges.

00As the same stream sometimes changes abruptly from one kind to the other, the two kinds must be due to different conditions affecting the flow, and among the conditions which may cause a stream to break up into the aa have been mentioned the greater depth of the stream, a sluggish current, impediments in its course just as it is granulating, and, what is more probable, subterranean moisture which causes it to cool from below upward instead of from above downward as in the pahoehoe.

00from Odense Fjord on the Odense Aa, the main portion on the north side of the stream, and the industrial Albani quarter on the south side.

00By the decree of the council of Trent he must be thirty Rom aa Y S' Cat h olic. years of age, of legitimate birth, and of approved learning and virtue.

00HERFORD, a town in the Prussian province of Westphalia, situated at the confluence of the Werre and Aa, on the Minden & Cologne railway, 9 m.

00The passage in Revelation runs: "Behold the tabernacle (o-Knvn) of God is with men, and He will tabernacle (vKnvc aa) with them."

00GRONINGEN, a town of Holland, capital of the province of the same name, at the confluence of the two canalized rivers the Drentsche Aa and the Hunse (which are continued to the Lauwers Zee as the Reit Diep), 16 m.

00The Aa church dates from 1465, but was founded in 1253.

00The Maas, whose course is almost parallel to that of the Rhine, follows in a wide curve the general slope of the country, receiving the Roer, the Mark and the Aa.

00If, ignoring temporarily and for simplicity the fact that part of the carbon may exist in the state of graphite, we consider the behaviour of iron in cooling from the molten state, AB and BC give the temperature at which, for any given percentage of carbon, solidification begins, and Aa, aB, and Bc that at which it ends.

00As the carbon-content increases the welding power naturally decreases rapidly, because of the rapid fall of the " solidus curve at which solidification is complete (Aa of fig.

00When the gradually falling temperature reaches 1430° (q), the mass begins to freeze as -y-iron or austenite, called " primary " to distinguish it from that which forms part of the eutectic. But the freezing, instead of completing itself at a fixed temperature as that of pure water does, continues until the temperature sinks to r on the line Aa.

00As freezing progresses, at each successive temperature reached the frozen austenite has the carbon-content of the point on Aa which that temperature abscissa cuts, and the still molten part or " mother-metal " has the carbon-content horizontally opposite this on the line AB.

00In other words, the composition of the frozen part and that of the mother-metal respectively are p and q at the beginning of the freezing, and r and t' at the end; and during freezing they slide along Aa and AB from p to r and from q to t'.

00That removal progressively raises the melting-point of the metal, after line Aa of fig.

0035, or by passing the piece through an aperture, F', smaller than the first F, as in case of the grooved rolls, AA, shown at the right, or by both means jointly.

00The Steenwyker Aa, however, enters the Zuider Zee independently.

00AA, Tower; B, ammoniacal carbonate, which brine main; E, gas-inlet; Z, vacuum filter; always contains some V, pipe to air pump.

00The Aµ from C to F being taken as unity in each case then the AA's for the other regions of the spectrum are expressed in fractions A A (C to F) and are given under the asterisks.

00AA, the name of a large number of small European rivers.

00a, aa, pronounced o).

00In Germany there are the Westphalian Aa, rising in the Teutoburger Wald, and joining the Werre at Herford, the Munster Aa, a tributary of the Ems, and others.

00If we write the expressions for the two lines in the form A+Bi, A'+B'i, the product is AA' - BB'+i(AB'+BA'); and the fact that the length of the product line is the product of those of the factors is seen in the form (A 2 + B 2) (A'2-{-B'2) = (AA' - BB')2+(AB'+BA')2.

00Let AB be the shortest distance between the lines of action, and let AA, BB

00C Let a, ~ be the angles which AA, BB make with the direction of the vector-sum, on opposite sides.

00AA.

00Resolving AA,

00It is evident, in the first place, that in any displacement common to the two surfaces, the work of the two equal and opposite normal pressures will cancel; moreover if, one of the surfaces being fixed, an infinitely small displacement shifts the point of contact from A to B, and if A be the new position of that point of the sliding body which was at A, the pro jectior of AA on the normal at A is of the second order.

00BA=7AC. Themasscentre (G) of a, $, y will then divide AA so that a .,AG = (t~+7) GA.

00If we imagine a point Q to describe a circle of radius a _________________ with the angular velocity ~, its A - 0 P orthogonal projection P on a fixed diameter AA will execute a vibration of this character.

00Also if ~ be the rate at which J describes the polhode, we have ~l sin (J3a) ~ sin (3, whence sin (aa)

00If 0 is constant the points C, A will in time at come to positions C, A such that CC = sin 0 i~, AA = cos 0 ~, and the angular momentum about OB will become Cn sin 0 ~ A sin 0 ~.

00(Cir aa,r)q, + (Cfr o2asr)qf +.

0088), where aa, bb, cc, dd represent plane a joints.

00Let the centre of R C pressure C at the first joint aa be known, and also the b a pressure P acting at C in C,

00The terrible power of excommunication is claimed for the church; but the council of the realm also is called to use the power given them by God to put down all religion but the reformed, and to further the aims and carry out the sentences AA.

00van der Aa, Biographisch Woordenboek der Nederlanden (Haarlem, 1851-1879); J.

00A piece of sheet metal is cut out in the form AA (fig.

00This film, however, tends to contract on itself, and the loose strip of metal BB will, if it is let go, be drawn up towards AA, provided it is sufficiently light and smooth.

00Hence if F is the force by which the slip BB is pulled towards AA, F =d b T ab =Ta,..

00who is a judge of lesser causes; the recorder (iEpop,v, iwv); and so on, down to the cleaners of the lamps (Aa ora86pwwL), the attendant of the lights and the bearer of the images (1 3ao - Tayap10s) and of the holy ointment (µvp056Tns).

00Other c= 1 Aa c>a FIG.

00Courland is drained by nearly one hundred rivers, of which only three, the Dvina, the Aa and the Windau, are navigable.

00In Aristotle, again, the principle which sets all nature under the rule of thought, and directs it towards a rational end, is vows, or the divine spirit itself; while Aoyos is a term with many senses, used as more or less identical with a number of phrases, ou €v€Ka, ivEpyaaa, ivr€X aa, ovwia, e hos, popcIA, &c.

00The other ridge of hills extends through the south-eastern division of the province called Twente, from Enschede to Ootmarsum, and divides the basin of the Almelosche Aa from the Dinkel and its streams. The river Vecht crosses the province from E.

00The mean sea-level is shown by the horizontal line aa, dotted where it passes through the land, and the natural mean level of saturation bb, above the sea-level, by a curved dot and dash line.

00These fissures are in vertical planes stretching entirely across the puddle trench, and reaching in one case, aa, nearly to the highest level at which the reservoir had been worked for seventeen years after the leakage had been discovered.

00(I-17) thrown into an anticline (AA') and syncline (BB').

00If a line S2 cut an arc aa at b, so that the two segments ab, ba lie on opposite sides of the line, then projecting the figure so that the line Sl goes off to infinity, the tangent at b is projected into the asymptote, and the arc ab is projected into a hyperbolic leg touching the asymptote at one extremity; the arc ba will at the same time be projected into a hyperbolic leg touching the same asymptote at the other extremity (and on the opposite side), but so that the two hyperbolic legs may or may not belong to one and the same branch.

00Next, if the line S2 touch at b the arc aa so that the two portions ab, ba lie on the same side of the line Sl, then projecting the figure as before, the tangent at b, that is, the line S2 itself, is projected to infinity; the arc ab is projected into a parabolic leg, and at the same time the arc ba is projected into a parabolic leg, having at infinity the same direction as the other leg, but so that the two legs may or may not belong to the same branch.

00SX may be also divided externally at A', so that SA'/A'X = e, since e is less than unity; the points A and A' are the vertices, and the line AA' the major axis of the curve.

00It is obvious that the curve is symmetrical about AA'.

00If AA' be bisected at C, and the line BCB' be drawn perpendicular to AA', then it is readily seen that the curve is symmetrical about this line also; since if we take S' on AA' so that S'A' =SA, and a.

00(a) Let AA', BB' be the axes intersecting at right angles in a point C. Take a strip of paper or rule and mark off from a point P, distances Pa and Pb equal respectively to CA and CB.

00(b) Let AA', BB' be the axes as before; describe on each as diameter a circle.

00Let AA' be the axis and S, S' the foci.

00Take a piece of thread of length AA', and fix it at its extremities by means of pins at the foci.

00For, draw through P a line parallel to AA', intersecting the directrix in Q and the line RSN in T.

00(4) If the axes be given, the curve can be approximately constructed by circular arcs in the following manner: - Let AA', BB' be the axes; determine D the intersection of lines through B and A parallel to the major and minor axes respectively.

00in extent) and on the river Aa.

00for a long distance, past Soleure (below which the Grosse Emme flows in on the right), Aarburg (where it is joined by the Wigger, right), Olten, Aarau, near which is the junction with the Suhr on the right, and Wildegg, where the Hallwiler Aa falls in on the right.

00Duro, Armada espanola, iv.; der Aa, Biograph.

00"Join AA," I answered.

00The AA is proud to have the benefit of an active and participatory democracy.

00I have aa 4-year degree in business and am a real estate investor.

00The third match, where a* matches aa, is interesting.

00Escape it the tote system or get a including the aa.

00Much less than least aa by is necessary or.

00As I said in my first post I'm really just looking for deals on bulk purchase of high mha aa and aaa batteries.

00aa rosette for food Set in the heart of the Cotswolds, Gloucestershire.

00aa breakdown Worried about breaking down at the roadside?

00aa battery [not included] .

00aa index are now close to those seen at the beginning of the dataset.

00aa o the tragedie.

00aa guns are far more common and more densely placed.

00Ay, it's spring again, WI aa the twists an turns that Mother Nature can throw at's.

00I hae the answer tae aa oor problems. ' ' Whit dae ye mean?

00The world o aa the senses is the same.

00Fit ill is there in a bawd, efter aa?

00Armament: Four 5.5 -inch guns Three 3.1 inch aa guns and Six 21 - inch torpedo tubes in three carrying 12 Torpedos.

00We war treated tae " Scots Wha Hae " an " A Man's a Man fur aa That " .

00Membership is open to all clubs affiliated to the South of England AA.

00Richard Archibald from Ballymoney Gled tae finally hear fae sae many Scotch folk that're aa fired up aboot thair ain leid.

00The shared experiences of active alcoholism at AA meetings make up a bond among all members of Alcoholics Anonymous.

00anaemiae who have AA blood can be assured that none of their children will develop sickle-cell anemia, no matter whom they marry.

00Map Map of Sicily, Italy [Based on AA Europe road Atlas, 1997] .

00By the time he got a call from an old drinking buddy in 1934, Wilson had, in AA parlance, hit bottom.

00build Rights Reserved | Crown Copyright This site has been built to comply with the W3C AA accessibility standards.

00An additional service that came into effect in 2002 involved hourly traffic bulletins with AA Roadwatch.

00The diagnosis of amyloidosis is histological, ie proof of AA amyloid deposition in biopsy material.

00Despite its rather downbeat image AA caters for all classes of society.

00The AA also provides a booklet for disabled drivers traveling in Europe, which is quite useful.

00AA personal finance Turnover in personal finance grew by 11% to £ 172 million.

00Aa wad be laith ti gan win him - wouldn't go with him.

00The company built a grandstand to seat 700, along with aa elliptical cycling track.

00hourly traffic bulletins with AA Roadwatch.

00AA honors vintage member A Suffolk pensioner has been handed honorary membership of the Automobile Association after her loyalty to the group.

00menadione sodium bisulfite) to... reported that AA plus vitamin K3 had growth inhibiting.. .

00Ben the pews the hum o spikk wis aa o the tragedie.

00orthopedics 1991;14:1253-1256. [PubMed Abstract] Agre JC, Rodriquez AA, Harmon RL, Swiggum ER, Franke TM.

00The third, and most interesting, group are those who are rescued by AA, going on to become paragons of sobriety.

00I checked last night on the AA site using their route planner to double check on our route over.

00As part of the campaign, the AA is trialing the distribution of high visibility, waterproof yellow ponchos among new and existing members.

00But he did not interview the alleged rapist - referred to as AA - or Prince Charles about the allegations.

00Many have spoken about how AA saved their lives since most people come to the organization when they have reached rock bottom.

00aa rosette for food Set in the heart of the Cotswolds, Gloucestershire.

00Results indicated a marginally significant difference in mathematics self-concept between the MD and AA groups but no difference between the AA and HA groups.

00sergeant in charge of an AA battery... .

00It holds up to four AA and two AAA batteries, and even allows simultaneous charging of the two types.

00He does he contain AA batteries Ages 8 and he does he arrive skein in china.

00They are all new and boxed Grade " AA " solid spruce top Flame Maple Back and Sides Rosewood fingerboard with 20 frets.

00All Rights Reserved | Crown Copyright This site has been built to comply with the W3C AA accessibility standards.

00swimwear range caters for AA to E cups.

0012-step fellowships borne out of Alcoholics Anonymous (AA ).

00AA use T3 (connected by walkways to T1 & T2 ).

00Between the valleys of the Aa and the Maas lies the long stretch of heavy high-fen called the Peel ("marshy land").

00In some old roofs the rafters are connected without any intervening ridge-plate, with the result that after Sectional elevation on AA.

00The five fixed webs are attached to the table which is secured to the bottom of the box by the screws p. The three movable webs are attached to the projections XX on the frame aa.

00The niche at the entrance, which is rarely wanting, served, no doubt, for the sentry on guard AA FIG.

00C AA.

00If a rectangle be constructed about AA' and BB', the diagonals of this figure are the "asymptotes" of the curve; they are the tangents from the centre, and hence touch the curve at infinity.

00Some properties of the curve will be briefly stated: If PN be the ordinate of the point P on the curve, AA' the vertices, X the meet of the directrix and axis and C the centre, then PN 2: AN.NA':: SX 2: AX.A'X, PN 2 is to AN.NA' in a constant ratio.

00The circle on AA' as diameter is called the auxiliarly circle; obviously AN.NA' equals the square of the tangent to this circle from N, and hence the ratio of PN to the tangent to the auxiliarly circle from N equals the ratio of the conjugate axis to the transverse.

00Aa 18 18

00Van der Aa, Biographisch woordenboek der Nederlanden (22 vols., Haarlem, 1852-1878).

00The southeastern corner of the province is traversed by the Westerwolde Aa, which discharges into the Dollart.

00The sides of the square are 6 in., and the centres AA are taken at 2 in.

00s o rtÃ‚°o - 1 Desiin??aa I w;A, Econfin r o i 0, ??

00'S Hertogenbosch ('sBosch, or den Bosch, French Bois-leDuc), the capital of the province of North Brabant, Holland, at the confluence of the rivers Dommel and Aa, which unite to form the Dieze, and a junction station 292 m.

00BREDA, a fortified town in the province of North Brabant, Holland, at the confluence of the canalized rivers Merk and Aa, 15 m.

00of Sa aa ?

00The minor Aik is aa, and is itself a determinant of order n-t.

00= 0; and such functions satisfy the differential equation aoaa i +2a0a 2 +3a 2 aa 3 +...

00+na n _ i aa n = 0.

00Such an expression as a l b 2 -a 2 b i, which is aa 2 ab 2 aa x 2 2 ax1' is usually written (ab) for brevity; in the same notation the determinant, whose rows are a l, a 2, a3; b2, b 2, b 3; c 1, c 2, c 3 respectively, is written (abc) and so on.

00A 1, A2 Ã¯¿½ Ai, A 1 A 2, A2 and then Ao = al Ai+2a1a2AIA2+a2 A2 - (a1A1+a2A2) 2 = a?, A l = (a 1 A 1 +a2A2) (alÃ¯¿½l +a2Ã¯¿½2) = aAaÃ¯¿½, A 2 = (alÃ¯¿½l +a2/-12) 2 = aM; so that A = aa l +2a A a u 152+aM5 2 = (aA6+a,e2)2; whence A1, A 2 become a A, a m, respectively and ?(S) = (a21+a,E2) 2; The practical result of the transformation is to change the umbrae a l, a 2 into the umbrae a s = a1A1 +a2A2, a Ã¯¿½ = a1/Ã¯¿½1 + a21=2 respectively.

00By similarly transforming the binary n ic form ay we find Ao = (aI A 1 +a2 A2) n = aAn A l = (alAi - I -a 2 A 2) n1 (a1Ã¯¿½1 +a2m2) = aa a Ã¯¿½ - A i n-1 A2, n-k k n-k k n-k k A = (al l+a2A2) (alÃ¯¿½1+a2Ã¯¿½2) = a A Ã¯¿½ =A 1 A2, so that the umbrae A1, A 2 are a A, a Ã¯¿½ respectively.

00Operation upon J results as follows D AA J = wJ; D A J=0; D Ã¯¿½A J =0;D Ã¯¿½Ã¯¿½ J = wJ.

00Now D A xA k = (n - k) A k; AÃ¯¿½ A k = k A?1; D Ã¯¿½A A k = (n - k) A k+1;D mÃ¯¿½ A k = kA k; (n - k)A ka - w Ak - 1 aA k = O; a _ J (n - k) A k +l A k = O; kA k Ak = wJ; equations which are valid when X 1, X 2, Ã¯¿½ 1, Ã¯¿½2 have arbitrary values, and therefore when the values are such that J =j, A k =akÃ¯¿½ Hence Ã‚°a-do +(n -1)71 (a2aa-+...

00= wj, aa 1 aa 2 a a 3 the complete system of equations satisfied by an invariant.

00If 0 be the degree of an invariant j - aj aj a; oj =a Ã‚° a a o +al aa l +...

00+anaan naj n.-1 aj naj =a l aa Ã‚° +a 1 a2c3a1...+a2aan and, herein transforming from a to x, we obtain the first evectant (-) k, x1x2 aak k Combinants.

00the two quadratic forms f, 4); the two discriminants (f, f')2,(0,4')2, and the first and second transvectants of f upon 4, (f,, >) 1 and (f, 402, which may be written (aa)a x a x and (aa) 2 .

00If a, a be the linear forms, above defined, he raises the identity ax(0) =ax(aJ3) - (3x(aa) to the fifth power (and in general to the power n) obtaining (aa) 5 f = (a13) 5 az - 5 (a0) 4 (aa) ax?3 -F...

00- (aa) and then expresses the coefficients, on the right, in terms of the fundamental invariants.

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.

00l aa k -x 2 d d- = 0; Z(nk)ak+l adk - x ldd2=0; or in the form d d 52-x 2(7 =0, O - x1ax2 = 0; where 0 = ao d a l + 2a 1 -?...+na,,_id an, 0 = nal dao -?

00then of course (AB) = (ab) the fundamental fact which appertains to the theory of the general linear substitution; now here we have additional and equally fundamental facts; for since A i = Xa i +,ia2, A2= - Ã¯¿½ay + X a2, AA =A?-}-A2= (X2 +M 2)(a i+ a z) =aa; A B =AjBi+A2B2= (X2 +, U2)(albi+a2b2) =ab; (XA) = X i A2 - X2 Ai = (Ax i + /-Lx2) (- /-jai + Xa2) - (- / J.x i '+' Axe) (X a i +%Ga^2) = (X2 +, u 2) (x a - = showing that, in the present theory, a a, a b, and (xa) possess the invariant property.

00(ab), aa, ab, (xa), ax, xx.

00By March 18 the Bolsheviks were thrown back over the Aa river.

00Thus (A +a) 2 = (A +a) (A+a) =A(A+a) +a(A+a)=AA+Aa+aA+aa; and the grouping of the second and third terms as 2Aa involves treating Aa and aA as identical.

00(b) In the same way, for the expansion of (Aa)", let o, denote uo-u 1 +.

00To apply the method to the calculation of N n, it is necessary that we should be able to express N in the form A+a or Aa, where a is small in comparison with A, A" is easy to calculate and a/A is convenient as a multiplier.

00Each symbol a is associated with its supplement a which satisfies the equivalences a+a = i, aa = o, the latter of which means that a and a have no region in common.

00A point A in space may be associated with a (real, positive, or negative) numerical quantity a, called its weight, and denoted by the symbol aA.

00The sum of two weighted points aA, (3B is, by definition, the point (a+13) G, where G divides AB so that AG: GB =0: a.

00X is, in general, a determinate point, the barycentre of aA, 3B, &c. (or of A, B, &c. for the weights a, 0, &c.).

00+X) happens to be zero, X lies at infinity in a determinate direction; unless - aA is the barycentre of (3B, yC, ...

00Then (de)(abc) = (abde)c+(cade)b+(bcde)a = (abce)d - (abcd)e, (ab) (AB) = (aA) (bB) - (aB) (bA) abic = (alc) b - (bjc)a, (ablcd) = (ajc) (bjd) - (af d) (bIc).

00The brightness (corresponding to B) midway between the two central points AA' is 8106 of the brightness at the central points themselves.

00If A' be the area over which the intensity must be Io in order to give the actual total intensity in accordance with A'102 =ff + 12d4dn, the relation between A and A' is AA' = X 2 f 2 .

00The position of Q corresponding to a given value of V, that is, to a band of given order, is by (19) BQ= aa b AD=V?

00HC:N ›NH N:CH ›NH HC:CH" Osotriazole (aa').

00Round the cylinder r=a held fixed in the U current the liquid streams past with velocity q' =2U sin 0+m/a; (2) and the loss of head due to this increase of velocity from U to q' is q' 2 -U 2 - (2U sin e to space filled with liquid, and at rest at infinity, the cylinder will experience components of force per unit length (i.) -27rpmV, 27rpmU, due to the vortex motion; 2 dU 2dV (ii.) -71-pa 2 w,, -7rpa dt, due to the kinetic reaction of the liquid; (iii.) o, -7r(a-p)a 2 g, due to gravity, taking Oy vertically upward, and denoting the density of the cylinder by a; so that the equations of motion are 71-0-a 2 - di r = - 7pa2- -- 22rpmV, (4) aa 2 - = -7rpa 2 dV +27rpmV - 7r(cr - p) a2g, (5) 7r or, putting m = a 2 w, so that the vortex velocity is due to an angular velocity w at a radius a, (o+p)dU/dt+2pwV =o, (6) (a+ p) dV /dt - 2 pwU + (v - p)g = o.

00If at any points of a vortex line the cross-section ABC, A'B'C' is drawn of the vortex filament, joined by the vortex line AA', then, since the flow in AA' is taken in opposite directions in the complete circuit ABC AA'B'C' A'A, the resultant flow in AA' cancels, and the circulation in ABC, A'B'C' is the same; this is expressed by saying that at all points of a vortex filament wa is constant where a is the cross-section of the filament and w the resultant spin (W.

00rf > a> b> o> a'> -Do; and then so that PT =c/Zir, and the curve AP is the tractrix; and the coefficient of contraction, or breadth of the jet breadth of the orifice - +i' A change of S2 and 0 into nS2 and nO will give the solution for two walls converging symmetrically to the orifice AA 1 at an angle zr/n.

00When the barrier AA' is held oblique to the current, the stream line xB is curved to the branch point B on AA' (fig.

00u -b' Along a jet surface, q=Q, and ch S2= cos 0 =cos a-i sin2a(a-a')/(u-b), (5) if 0 =-a at the source x of the jet xB, where u = co; and supposing 0=0,13 at the end of the streams where u =j, j', u-b i sin 2 a u - j cos 0-cos /3 i a -a cos a sin a -cos 0' aa' - 2 (cos a -cos (3) (cos a-cos 0)' u-j' 1 2 cos 0-cos, (6) a -a' - 2 S i n a (cos a -cos (3') (cos a -cos B)' and 4' being constant along a stream line d4 - dw ds _d8 d4 _ dw du du du' d- -dud0' 7rQ ds_ it ds (cos a-cos /3) (cos a -cos (3') sin 0 m+m' dB c d0 - (cos a-cos B) (cos 0-cos /3) (cos 0 -cos /3')' _ sin 0 cos a-cos 13 sin 0 - cos a-cos B + cos 0-cos (3' cos 0-cos 13 cos a -cos $ sin 6 cos (3-cos /3' cos 0-cos 0" giving the intrinsic equation of the surface of a jet, with proper attention to the sign.

00I, Aa) which is regarded as a sucker or as re lated to the uterine :; o o??

00______ Pilgrim and Trade Routes Swamps EH B Aa Mediterranean Muwdla sko / - /2nd.

00The harbour is connected by canals with the river Aa and the navigable waterways of the department.

00and 655,800 inhabitants) (5) Parts of the former province of Courland between the old German frontier and the Holy Aa river, as also part of the district of Illuxt.

00Cramer, light has been thrown on the period by the researches of de Hoop Scheffer; see Van der Aa, Biographisch woordenboek der Nederlanden (1869) R.

00The classification of the different kinds of coal may be considered from various points of view, such as their chemical composition, their behaviour when subjected to heat aa s sifica= or when burnt, and their geological position and iron.

00(iii) Take any straight line intersecting or not intersecting the figure, and draw perpendiculars Aa, Bb, Cc, Dd,.

00Its surface is diversified by several plateaus, those of Haanhof and of the Livonian Aa having an average elevation of 400 to 700 ft., while several summits reach Boo to loon ft.

00The edges of the plateaus are gapped by deep valleys; the hilly tract between the Dvina and its tributary the Livonian Aa has received, from its picturesque narrow valleys, thick forests and numerous lakes, the name of "Livonian Switzerland."

00along its frontier, the Pernau, Salis, Livonian Aa and Embach are navigable.

00Live-stock and dairy products are important factors in the T C t o L %i, aa.

00on in time dt to A'B'C', where AA' = Udt, the displacement of the particle originally at M must change from PM to P'M or by PP'.

00Now taking equation (72), and replacing tan B, as a variable final tangent of an angle, by tan i or dyldx, (75) tan 4) - dam= C sec n [I(U) - I(u)], and integrating with respect to x over the arc considered, (76) x tan 4, - y = C sec n (U) - f :I(u)dx] 0 But f (u)dx= f 1(u) du = C cos n f x I (u) u du g f() =C cos n [A(U) - A(u)] in Siacci's notation; so that the altitude-function A must be calculated by summation from the finite difference AA, where (78) AA = I (u) 9 = I (u) or else by an integration when it is legitimate to assume that f(v) =v m lk in an interval of velocity in which m may be supposed constant.

005, Aa and H).

00aa, ae, af, ag, ah, ak, al, am, an, ap, aq, ar, as, at, au, av, aw, ax, ay, az, bb, In Anodonta these pallial tentacles are confined to a small area surrounding the inferior siphonal notch (fig.

00i (5), one of the labial tentacles n is also thrown back to show the mouth w, and the two left gill-plates are reflected to show the gill-plates of the right side (rr, rq) projecting behind the foot, the inner or median plate of each side being united by concrescence to its fellow of the opposite side along a continuous line (aa).

00374: "AA represents a small scale.

004, when it ought properly to sink almost to the top of the stem, the surface tension of the water around the circumference of the circle of contact, AA', providing the additional support required.

00The three chief rivers of the province are the Elbe in the north-east, where it mainly forms the boundary and receives the navigable tributaries Jeetze, Ilmenau, Seve, Este, Luhe, Schwinge and Medem; the Weser in the centre, with its important tributary the Aller (navigable from Celle downwards); and in the west the Ems, with its tributaries the Aa and the Leda.

00van der Aa, Biog.

00The surface of the upper slopes of Mauna Loa is almost wholly of two widely different kinds of barren lava-flows, called by the Hawaiians the pahoehoe and the aa.

00The aa is lava broken into fragments having sharp and jagged edges.

00As the same stream sometimes changes abruptly from one kind to the other, the two kinds must be due to different conditions affecting the flow, and among the conditions which may cause a stream to break up into the aa have been mentioned the greater depth of the stream, a sluggish current, impediments in its course just as it is granulating, and, what is more probable, subterranean moisture which causes it to cool from below upward instead of from above downward as in the pahoehoe.

00

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