# How to use *Ab* in a sentence

In nonflowering plants the works usually followed are for ferns, Hooker and Baker's Synopsis filicum; for mosses, Muller's Synopsis muscorum frondosorum, Jaeger & Sauerbeck's Genera et species muscorum, and Engler & Prantl's Pflanzenfamilien; for algae, de Toni's Sylloge algarum; for hepaticae, Gottsche, Lindenberg and Nees

**ab**Esenbeck's Synopsis hepaticarum, supplemented by Stephani's Species hepaticarum; for fungi, Saccardo's Sylloge fungorum, and for mycetozoa Lister's monograph of the group. For the members of large genera, e.g.A young senator (

**ab**actis senatus) was chosen to draw up these Acta, which were kept in the imperial archives and public libraries.To explain this result, chemists suppose that both changes can occur simultaneously, and that equilibrium results when the rate at which

**AB**and CD are transformed into AD and CB is the same as the rate at which the reverse change goes on.Thus, if the molecule of a substance in solution is represented by

**AB**, Grotthus considered a chain of**AB**molecules to exist from one electrode to the other.Let

**AB**be the major axis of the orbit, B the pericentre, F the focus or centre of motion, P the position of the body.AdvertisementSuch 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.This will be recognized as the resultant of the two linear forms. If the two linear forms be identical, the umbral sets a l, a2; b l, b 2 are alternative, are ultimately put equal to one another and (

**ab**) vanishes.For the cubic (

**ab**) 2 axbx is a covariant because each symbol a, b occurs three times; we can first of all find its real expression as a simultaneous covariant of two cubics, and then, by supposing the two cubics to merge into identity, find the expression of the quadratic covariant, of the single cubic, commonly known as the Hessian.From the three equations ax = alxl+ a2x2, b.= blxl+b2x2, cx = clxi+c2x2, we find by eliminating x, and x 2 the relation a x (bc)+b x (ca) +c x (

**ab**) =0.The identities are, in particular, of service in reducing symbolic products to standard forms. A symbolical expression may be always so transformed that the power of any determinant factor (

**ab**) is even.AdvertisementFor two factors the standard form is (

**ab**) 2; for three factors (**ab**) 2 (ac); for four factors (**ab**) 4 and (**ab**) 2 (cd) 2; for five factors (**ab**) 4 (ac) and (**ab**) 2 (ac)(de) 2; for six factors (**ab**) 6, (**ab**) 2 (bc) 2 (ca) 2, and (**ab**) 2 (cd) 2 (ef) 2 .For the quadratic it is the discriminant (

**ab**) 2 and for ax2 the cubic the quadratic covariant (**ab**) 2 axbx.If we have a symbolic product, which contains the symbol a only in determinant factors such as (

**ab**), we may write x 2, -x 1 for a 1, a 2, and thus obtain a product in which (**ab**) is replaced by b x, (ac) by c x and so on.The first transvectant, (f,f') 1 = (

**ab**) a x b x, vanishes identically.This can be verified by equating to zero the five coefficients of the Hessian (

**ab**) 2 axb2.AdvertisementSolving 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.In the a x = aixi+a2x2, observe that a a = a2,

**ab**= aibi +a2b2.The general form of covariant is therefore (

**ab**) h i (ac) h2 c) (b h3 a i bb2c'e3...abia?2b?3...To assist us in handling the symbolic products we have not only the identity (

**ab**) cx + (bc) a x + (ca) bx =0, but also (**ab**) x x+ (b x) a + (ax) b x = 0, (**ab**)a+(bc)a s +(ca)a b = 0, and many others which may be derived from these in the manner which will be familiar to students of the works of Aronhold, Clebsch and Gordan.Let the distance of each pole of the rod

**AB**from the centre of the magnetometer needle = d.AdvertisementDenoting the distance AM by d 1, BM by d2, and

**AB**by 1, we have for the force at M due to the magnetism of the rod H P =d 12 - horizontal component (dla - d 2 3).The city then placed itself under the protection of the emperor .Charles V., created a magistracy of "ten conservators of the liberties of the state" (December 1524), united the different monti in one named the "monte of the reigning nobles," and, rejoicing to be rid of the last of the Petrucci, dated their public books,

**ab**instaurata libertate year I., II., and so on.Thus a(b+c) and (b+c)a give the same result, though it may be written in various ways, such as abdac, ca+

**ab**, &c. In the same way the associative law is that A(BC) and (**AB**)C give the same formal result.Let a+b denote the region made up of a and b together (the common part, if any, being reckoned only once), and let a X b or

**ab**mean the region common to a and b.It is remarkable that Mobius employs the symbols

**AB**, ABC, Abcd In Their Ordinary Geometrical Sense As Lengths, Areas And Volumes, Except That He Distinguishes Their Sign; Thus**Ab**= Ba, Abc= Acb, And So On.AdvertisementThe symbol

**AB**, from this point of view, is called the inner product of A and B, as distinguished from the outer product IAB.In the course of reducing such expressions as (

**AB**)C, (**AB**){C(DE)} and the like, where a chain of multiplications has to be performed in a certain order, the multiplications may be all progressive, or all regressive, or partly, one, partly the other.In ordinary algebra we have the disjunctive law that if

**ab**= o, then either a = o or b= o.By the principle of energy the illumination over the entire focal plane must be equal to that over the diffracting area; and thus, in accordance with the suppositions by which (3) was obtained, its value when integrated from E= co to = -1-x, and from n = - oo to n = -1-oo should be equal to

**ab**.Let AoBo be a plane wave-surface of the light before it falls upon the prisms,

**AB**the corresponding wave-surface for a particular part of the spectrum after the light has passed the prisms, or after it has passed the eye-piece of the observing telescope.The path of a ray from the wave-surface AoBo to A or B is determined by the con dition that the optical distance, µ ds, is a minimum; and, as

**AB**is by supposition a wave-surface, this optical distance is the same for both points.B) - f S,u ds (along The new wave-surface is formed in such a position that the optical distance is constant; and therefore the dispersion, or the angle through which the wave-surface is turned by the change of refrangibility, is found simply by dividing (5) by the distance

**AB**.If now we suppose the aperture

**AB**to be covered by a great number of opaque strips or bars of width d, separated by transparent intervals of width a, the condition of things in the directions just spoken of is not materially changed.In another direction, making a small angle with the last, such that the projection of

**AB**upon it amounts to a few wavelengths, it is easy to see that the mode of interference is the same as if there were no ruling.For example, when the direction is such that the projection of

**AB**upon it amounts to one wave-length, the elementary components neutralize one another, because their phases are distributed symmetrically, though discontinuously, round the entire period.Let P, Q denote the normal thrust across the sides bc, ca, and R the normal thrust across the base

**ab**.This is proved by taking any two points A and B at the same level, and considering the equilibrium of a thin prism of liquid

**AB**, bounded by planes at A and B perpendicular to**AB**.As gravity and the fluid pressure on the sides of the prism act at right angles to

**AB**, the equilibrium requires the equality of thrust on the ends A and B; and as the areas are equal, the pressure must be equal at A and B; and so the pressure is the same at all points in the same horizontal plane.Sargon, who meanwhile had crushed the confederacy of the northern nations, had taken (717 B.C.) the Hittite stronghold of Carchemish and had annexed the future kingdom of Ecbatana, was now accepted as king by the Babylonian priests and his claim to be the successor of Sargon of Akkad acknowledged up to the time of his murder in 705 B.C. His son Sennacherib, who succeeded Serena- hi m on the 12th of

**Ab**, did not possess the military or cherlb. ?Another castle, built in the same century, on the east bank, was held direct by the lords of Glamorgan, as the westernmost outpost of their lordship. It was frequently attacked by the Welsh, notably in 1231 when it was taken, and the town demolished by Llewelyn

**ab**Iorwerth.Radii are drawn from the centre of the quadrant to the points of division of the arc, and these radii are intersected by the lines drawn parallel to BC and through the corresponding points on the radius

**AB**.The castle was begun, in or about 1109, by Cadwgan

**ab**Bleddyn**ab**Cynfyn (Cynvyn), and finished by Gwenwynwyn; in 1196 it was besieged, undermined and taken by Hubert, archbishop of Canterbury.The founda 1 It must be remembered that the Pisans and Florentines dated the beginning of the year

**ab**incarnatione, i.e.Laennec, to whom we are indebted for the practice of auscultation, freely admits that the idea was suggested to him by study of Hippocrates, who, treating of the presence of morbid fluids in the thorax, gives very particular directions, by 1 " Hippocrates Cous, primus quidem ex omnibus memoria dignus,

**ab**studio sapientiae disciplinam hanc separavit, vir et arte et facundia insignis " (Celsus, De medicina).One of the best methods for doing this is to charge the

**Ab**l condenser by the known voltage of a battery, and then d e t erdischarge it through a galvanometer and repeat this minations.Then the potential at any point P in this ideal plane PO is equal to q/AP-q/BP=0, whilst the resultant force at P due to the two point charges is 2gAO/AP 3, and is parallel to

**AB**or normal to PO.A cycle such as ABCD enclosed by parts of two isothermals, BC, AD, and two adiabatics,

**AB**, CD, is the simplest form of cycle for theoretical purposes, since all the heat absorbed, H', is taken in during the process represented by one isothermal at the temperature o', and all the heat rejected, H", is given out during the process represented by the other at the temperature 0".It is not necessary in this example that

**AB**, CD should be adiabatics, because the change of volume BC is finite.If we consider any short length of the stream bounded by two imaginary cross-sections A and B on either side of the plug, unit mass of the fluid in passing A has work, p'v', done on it by the fluid behind and carries its energy, E'+ U', with it into the space

**AB**, where U' is the kinetic energy of flow.In passing B it does work, p"v", on the fluid in front, and carries its energy, E"+ U", with it out of the space

**AB**.If there is no external loss or gain of heat through the walls of the pipe, and if the flow is steady, so that energy is not accumulating in the space

**AB**, we must evidently have the condition E'+U'+p'v' =E'+ U"+p"v" at any two cross-sections of the stream.Meanwhile Aquitaine was gradually lost; the defeat of Pembroke off La Rochelle deprived England of the command of the sea, and Sir Owen ap Thomas, a grand-nephew of Llewelyn

**ab**Gruffyd, planned, with French help, an abortive invasion of Wales.Then MA'B'N is a right trapezium, whose area is equal to that of Cabd; and it is related to the latter in such a way that, if any two lines parallel to AC and BD meet

**AB**, CD, MN, A'B', in E, G, P, E', and F, H, Q, F', respectively, the area of the piece PE'F'Q of the right trapezium 'B.Then it may be shown that DE is parallel to

**AB**, and that the area of the figure between chord DE and arc DE is half the sum of the areas DHA and EJB.For a tetrahedron, two of whose opposite edges are

**AB**and CD, we require the area of the section by a plane parallel to**AB**and CD.Let the distance between the parallel planes through

**AB**and CD be h, and let a plane at distance x from the plane through**AB**cut the edges AC, up -f- .As

**AB**moves from CD to EF it pushes forward the layer of air in contact with it.As

**AB**returns from EF towards CD the layer of air next to it follows it as if it D E F were pulled back by**AB**.On the whole the air S within ABCD neither gains nor g D loses momentum, so that on the whole it receives as much through

**AB**as it gives up to CD.If P is the undisturbed pressure and P+w the pressure at

**AB**, the momentum entering through**AB**per second isJ01(P+w-+pu2)dt.But the values of 2 which occur successively during the second at

**AB**exist simultaneously at the beginning of the second over the distance U behind**AB**.Or if the conditions along this distance U could be maintained constant, and we could travel back along it uniformly in one second, we should meet all the conditions actually arriving at

**AB**and at the same intervals.If the train of waves is reflected, the value of p at

**AB**will be the sum of the values for the two trains, and will, on the average, be doubled.Then move

**AB**from right to left with this velocity, and the disturbance remains fixed in space.Since the conditions in the region PQ remain always the same, the momentum perpendicular to

**AB**entering the region at Q is equal to the momentum perpendicular to**AB**leaving the region at P. But, since the motion at Q is along**AB**, there is no momentum there perpendicular to**AB**.The material between A and B, though continually changing, is always in the same condition, and therefore the momentum within

**AB**is constant.Set off these distances along

**AB**and let Y1, y2,.Thus we shall call the first inclined line on the left hand the line AG, the line representing the first force on the top left-hand joint

**AB**, the first horizontal member at the top left hand the line BH, &c; similarly each point requires at least three letters to denote it; the top first left-hand joint may be called Abhg, being the point where these four spaces meet.All real connexion or relation, therefore, and with it all possibility of an objective system, disappears; it is, in fact, excluded by Hume

**ab**initio, for " the mind never perceives any real connexion among distinct existences."There is no evidence of any settlement on the site of the present town prior to about 1092, when Bernard Newmarch, after defeating Bleddin

**ab**Maenarch, built here a castle which he made his residence and the chief stronghold of his new lordship. For this purpose he utilized what remained of the materials of the Roman fort, 3 m.The second class, referred to in inscriptions by the name of praefecti

**ab**decurionibus creati lege Petronia, seem to have been appointed by the local senate in case of a complete absence of higher magistrates, such as would have led in Rome to the appointment of an interrex.The line

**AB**is termed the " radical axis."To prove this let

**AB**,**AB**' be the tangents from any point on the line AX.Since PD = PD', it follows that

**AB**=**AB**'.Take

**AB**equal to one-fourth of the given line; on**AB**describe a square ABCD; join AC; in AC produced find, by a known process, a point C 1 such that, when C 1 B 1 is drawn perpendicular to**AB**produced and C 1 D 1 perpendicular to BC produced, the rectangle BC,.The diameter sought is the straight line from A to the limiting position of the series of B's, say the straight line

**AB**co.Denoting the number of units in

**AB**by 4c, we can express BB 1j B 1 B 2,.The town derives its name from the river Avon (corrupted from Avan), which also gave its name to a medieval lordship. On the Norman conquest at Glamorgan, Caradoc, the eldest son of the defeated prince, Lestyn

**ab**Gwrgan, continued to hold this lordship, and for the defence of the passage of the river built here a castle whose foundations are still traceable in a field near the churchyard.He also brought out editions of the Byzantine historians, John Cinnamus and John Zonaras, as Joannis Cinnami historiarum de rebus gestis a Joanne et Manuele Comnenis (Paris, 1670) and Joannis Zonarae Annales

**ab**exordio mundi ad mortem Alexii Comneni (Paris, 1686).The diagonal AD of the parallelogram, of which

**AB**and AC are adjacent sides, will represent, both in direction and magnitude, the motion of the rain as apparent to the observer.At A we 66 have the freezing point of pure water, which is lowered by the gradual addition of 46 ferric chloride in the manner shown by the curve

**AB**.To explain the universe Spinoza proceeds to argue that God, though undetermined

**ab**extra, is capable of infinite self-determination.If, 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.As 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**.In 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'.Thus, we find Sumerian

**ab**, " dwelling," " sea ";**ab**, " road," and -**ab**, a grammatical suffix, which words, with many others of a similar character, were perhaps originally uttered with different voice-tones.In the logic of Mill, e.g., we find much of a special character that has no counterpart in Hume, much that is introduced

**ab**extra, from general considerations of scientific procedure, but, so far as the groundwork is concerned, the System of Logic is a mere reproduction of Hume's doctrine of knowledge.Working on these lines, and attaching special importance to common descent, naturalists applied the term with more and more precision, until Linnaeus, in his Philosophia botanica, gave the aphorism, "species tot sunt diversae, quot diversae formae

**ab**initio sunt creatae" - "just so many species are to be reckoned as there were forms created at the beginning."We thus obtain the differential equation gk(d 2 0/dx 2) =cgdo/dt+hpo, which is satisfied by terms of the type =c" sin where a 2 -b 2 = hp/qk, and

**ab**= urnc/k.He was also the author of volumes entitled De satisfactione Christi disputationes (Geneva, 1666) and De necessaria secessione nostra

**ab**ecclesia romana (Geneva, 1687).Bernard himself initiated this policy by building a castle at Talgarth on the Upper Wye, but in 1091 he moved southwards, defeated the regulus of Brycheiniog, Bleddyn

**ab**Maenarch, and his brother-in-law Rhys ap Tewdwr, the prince of south-west Wales, and with materials obtained from the Roman fort of Caer Bannau, built a castle at Brecon, which he made his caput baroniae.Then if a pencil be placed along B C so as to keep the string taut, and the limb

**AB**be slid along the directrix, the A pencil will trace out the parabola.His orations and letters were published in 1492; but his title to any measure of fame he possesses rests upon his history of Venice, De origine urbis Venetiarum rebusque

**ab**ipsa gestis historia (1492), which was translated into Italian by Domenichi in 1545, and which at the time of its appearance was undoubtedly the best work upon the subject of which it treated.The axis of rotation

**AB**bears a rigidly attached rod DBC inclined to it at an angle equal to the sun's polar distance.A large fire-grate (

**ab**), having a cave (c) to facilitate stoking and stepped back at (d), is bounded on one side by a fire-bridge (e); on the other side of this, separated by an air-channel (g), there is first the proper fluxing bed (h), and behind this the "back-bed "(i) for pre-heating the charge.Thus the figure represents a section the (ideally simplified) uni verse cut perpendicular to C P' D the planes

**AB**and CD between which the stars are contained, 1 This number is the 3/2th power of the ratio of the brightness of stars differing by a unit magnitude.Imagine this stratum to be uniformly filled with stars (of course in the actual universe instead of sharply defined boundaries

**AB**and CD, we shall have a gradual thinning out of the stars) it follows that in the two directions SP and SP' the fewest stars will be seen; these then are the directions of the galactic poles.An army sent to the rescue by Ibn Zobair under the command of his brother Mus`

**ab**was beaten in Palestine by `Amr Ashdaq.Mokhtar was now at the zenith of power, but Ibn Zobair, determined to get rid at all costs of so dangerous an enemy, named his brother Mus`

**ab**governor of Basra and ordered him to march against Kufa.Mohallab then marched with Mus`

**ab**against Kufa.Ibn Zobair, however, was occupied at Mecca with the rebuilding of the Ka`ba, and Mus`

**ab**was harassed not only by the Kharijites, but also by a noble freebooter, Obaidallah b.When, in the year (69 A.H.) 689 Abdalmalik had at last encamped at Botnan Habib in the vicinity of Kinnesrin (Qinnasrin),1 with the purpose of marching against Mus`

**ab**, his cousin `Amr Ashdaq, to whom by the treaty of Jabia, before the battle of Merj Rahit, the succession to Merwan had been promised, took advantage of his absence to lay claim to the supreme power, and to have himself proclaimed caliph by his partisans.Mus`

**ab**was encamped at Bajomaira in the neighbourhood of Takrit.Meanwhile, Mus`

**ab**had to curb a violent revolt in Basra, brought about by agents of Abdalmalik, and called after a place in the city the revolt of the Jofrites.Mus`

**ab**'s best troops were fighting under Mohallab against the Kharijites; many Basrians were secretly favourable to the Omayyads, nor were the Kufian soldiers to be trusted.Abdalmalik, therefore, wrote secretly to the chiefs of Mus`

**ab**'s army, and persuaded them to desert to him, with the exception of Ibrahim b.His death, in the beginning of the battle, decided the fate of Mus`

**ab**, who was slain sword in hand by a Shiite of Kuf a.As soon as the expedition to Irak against Mus`

**ab**had terminated, the holy war against the Greeks was renewed.That the most important section, the list of forms of combination, was never achieved - this too was after the Baconian example while the mode of symbolization was crude with a=

**ab**and the like - matters little.Tradition ascribes the death of Aaron to the first day of

**Ab**.It is therefore adequately represented, for mathematical purposes, by a straight line

**AB**drawn in the direction in question, of length proportional (on any convenient scale) to the magnitude of the force.Since the four co-ordinates (Cartesian or other) of these two points are connected by the relation which expresses the invariability of the length

**AB**, it is plain that virtually three inde pendent elements are re quired and suffice to specify the position of the lamina.For suppose that in consequence of the displacement a point of the lamina is brought from A to B, whilst the point of the lamina which was originally at B is brought to C. Since

**AB**, BC, are two different positions of the same line in the B C lamina they are equal, and it is evident that the rotation could have been effected by a rotation about J,As a special case A the three points A, B, C may be in a straight line; J is then at infinity and the displacement is equivalent to FIG 10 a pure translation, since every point of the lamina is now displaced parallel to

**AB**through a space equal to**AB**.Hence the locus of J relative to

**AB**, and the locus relative to CD are equal ellipses of which A, B and C, D are respectively the foci.These formulae give a means of constructing the resultant by means of any transversal

**AB**cutting the lines of action.When P, Q have opposite signs the point C divides

**AB**externally on the side of the greater force.Again, suppose we have a bar

**AB**resting with its ends on twc smooth inclined planes which face each other.If F, Q, R, be any three forces acting along BC, CA,

**AB**, respectively, the line of action of the resultant is determined by the consideration that the sum of the moments about any point on it must vanish.Thus in the figure the sum of the moments about 0 is P. OAP. OB or P.

**AB**, which is independent of the position of o 0.The force corresponding to

**AB**may be replaced by the two components marked x, y; and a force corresponding to BA may be represented by the two components marked x, y.Thus ii

**AB**, BC, CD represent the given loads, in the force-diagram, we construct the sides corresponding to OA, OB, OC, OD in the funicular; we then draw the closing line of the funicular polygon, and a parallel OE to it in the force diagram.A graphical method can also be applied to find the moment of a force, or of a system of forces, about any assigned point P. Let F be a force represented by

**AB**in the force-diagram.To find the pressure exerted by a bar

**AB**on the pin A we compound with the force in**AB**given by the diagram a force equal to P. Conversely, to find the pressure of the pin A on the bar**AB**we must compound with the force given by the diagram a force equal and opposite to P. This question arises in practice in the theory of three-jointed structures; for the purpose in hand such a structure is sufficiently represented by two bars**AB**, BC. The right-hand figure represents a portion of the force-diagram; in particular ZX represents the pressure of**AB**on BThe spherical A isosceles triangles AJB, BJC are con gruent, and we see that

**AB**can be brought into the position BC by a rotation about the axis OJ through an FIG.This is the axis of the required screw; the amount of the translation is measured by the projection of

**AB**or BC or CD on the axis; and the angle of rotation is given, by the inclination of the aforesaid bisectors.An infinitely small rotation about any axis is conveniently represented geometrically by a length

**AB**measures along the axis and proportional to the angle of rotation, with the convention that the direction from A to B shall be related to the rotation as is the direction of translation to that of rotation in a righthanded screw.If

**AB**, AC represent infinitesimal rotations about intersecting axes, the consequent displacement of any point 0 in the plane BAC will be at right angles to this plane, and will be represented by twice the sum of the areas OAB, OAC, taken with proper signs.Let

**AB**be the shortest distance between the lines of action, and let AA, BBThis product is the same whatever point on the line of action be taken, since the lengthwise components of the dispJacements of any two points A, B on a line

**AB**are equal, to the first order of small quantities.It is to be noticed that All need not be in the same plane with

**AB**, AC.Imagine the bar CF to be removed, and consider a deformation in which

**AB**is fixed.The simplest case is that of a frame of three bars, when the three joints A, B, C fall into a straght line; a small displacement of the joint B at right angles to AC would involve changes in the lengths of

**AB**, BC which are only of the second order of small quantities.If E be the point in which the line of the string meets

**AB**, we have pjCP, pz=EP. Many contrivances for actually drawing the resulting curves have been devised.The line T on the surface bbb has for the instant no velocity it a direction perpendicular to

**AB**; becau2e for the instant it touches, without sliding, the line T on the fixed surface aaa.The line T on the surface bbb has also for the instant no velocity in the plane

**AB**; for it has just ceased to move towards the fixed surface aaa, and is just about to begin to move away from that surface.Set off

**ab**= ac = 1/2p. Draw radii bd, Ce; draw fb, cg, making angles of e 753/4 with those radii.The links

**AB**and CD are equal in length and are centred respectively at A and C. The ends D and B are joined by a link DB.But tile point B must also move at right angles to

**AB**in the case under consideration.Hence draw a line through O in the velocity diagram at right angles to

**AB**to cut cbi in b.The year of

**Ab**Saids death is most likely that of the first great didactic mathnawi, the Rshan.There exists a mid14th-century Historia do Santo Graal, and an unprinted Josep'

**ab**Aramadia, while, though the MS. is lost, we have abundant evidence of the existence of a primitive Portuguese prose redaction of Amadis de Gaula anterior to the present Spanish text.If, in the rectangle Aacc, Aa= a, and AC = b, its area is S =

**ab**, and its energy Tab.Hence if F is the force by which the slip BB is pulled towards AA, F =d b T

**ab**=Ta,..For if in the triangle abc the side

**ab**is taken so as to represent on a given scale the tension of the surface of contact of the fluids a and b, and if the other sides be and ca are taken so as to represent on the same scale the tensions of the surfaces between b and c and between c and a respectively, then the condition of equilibrium at 0 for the corresponding tensions R, P and Q is that the angle ROP shall be the supplement of abc, POQ of bca, and, therefore, QOR of cab.If four fluids, a, b, c, d, meet in a point 0, and if a tetrahedron

**AB**CD is formed so that its edge**AB**represents the tension of the surface of contact of the liquids a and b, BC that of b and c, and so on; then if we place this tetrahedron so that the face ABC is normal to the tangent at 0 to the line of concourse of the fluids abc, and turn it so that the edge**AB**is normal to the tangent plane at 0 to the surface of contact of the fluids a and b, then the other three faces of the tetrahedron will be normal to the tangents at 0 to the other three lines of concourse of the liquids, an the other five edges of the tetrahedron will be normal to the tangent planes at 0 to the other five surfaces of contact.But once the idea of supernatural grace going along with office as such (of which we have already a trace in the Ignatian bishop, though without the notion of actual apostolic succession) arose in connexion with successio

**ab**apostolis, the full development of the doctrine was but a matter of time.4 Literature.Thus Herbert sought to do for the religion of nature what his friend Grotius was doing for natural law, - making a new application of the standard of Vincent of Lerins, Quod semper, quod ubique, quod

**ab**omnibus.Its proper title was

**Ab**urbe condita libri (also called historiae and annales).Over a mile south of Mold, on the right of the road to Nerquis, is the "Tower" (15th century, but perhaps restored in the 18th), where, in 1465 or 1 475, the royal chieftain, Rheinallt

**ab**Gruffyd ad Bleddyn, hanged Robert Byrne, mayor of Chester, and subsequently burned alive some 200 Chester folk who tried to 'arrest him.Draw any line DE perpendicular to

**AB**and meeting the circle in E, and take a point P on DE such that the line DP =arc BE; then the locus of P is the companion to the cycloid.Not, indeed, a preliminary criticism of our faculties or conceptions such as Kant himself proposed to institute, in order to determine the limits of their application; such a criticism

**ab**extra of the nature of our experience is essentially a thing impossible.Among his other works are his Annotationes in Biblia (1607), of which an English translation (Pious and Learned Annotations upon the Holy Bible) was published in London in 1648, and various polemical treatises, such as De fictitio Pontificiorum Purgatorio (1619); De justa secessione Reformatorum

**ab**Ecclesia Romana (1628); De Antichristo, &c. He also published French translations of Sarpi's History of the Council of Trent, and of Edwin Sandys's Account of the State of Religion in the West.There is considerable distortion of the clay, resulting from combined shearing and tensile stress, above each of the steps of rock, and reaching its maximum at and above the highest rise

**ab**, where it has proved sufficient to produce a dangerous line of weakness ac, the tension at a either causing actual rupture, or such increased porosity as to permit of percolation capable of keeping open the wound.It is clear that the greatest crushing pressure must occur, either, with the reservoir empty, near the lower part of the water face

**ab**.We have hitherto considered only the horizontal overturning pressure of the water; but if from originally defective construction, or from the absence of vertical pressure due to weight of masonry towards the water edge of any horizontal bed, as at

**ab**in fig.Line

**ab**is in such a position that the stresses.The vertical distance above the line

**ab**of any point in the dotted line dc is proportional to the vertical component of the compressive stress on the line ale assumed to vary uniformly from face to face, and similarly the vertical distance of any point in the3-dot-and-dash line ae above the line**ab**is proportional to the vertical component of the stress determined experimentally.In order to increase the sensitiveness of a balance, the line

**AB**joining the points of suspension and the centre of gravity of the balance must be brought nearer to each other.Io

**AB**is the steelyard.The cartesian equation, if A be taken as origin and

**AB**(= 2a) for the axis of x, is xy 2 =4a2(2a - x).If 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.Next, 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.To this new period belong Einleitung zur historic der vornehmsten Reiche and Staaten, also the Commentarium de rebus suecicis, libri XXVI.,

**ab**expeditione Gustavi Adolphi regis in Germaniam ad abdicationem usque Christinae and De rebus a Carolo Gustavo gestis.Bellarmine (De jejunio) distinguishes jejunium spirituale (abstinentia a vitiis), jejunium morale (parsimonia et temperantia cibi et potus), jejunium naturale (abstinentia

**ab**omni prorsus cibo et potu, quacunque ratione sumpto), and jejunium ecclesiasticum.Thus, for three columns, it appears by either rule that 123, 231, 312 are positive; 213, 321, 132 are negative; and the developed expression of the foregoing determinant of the third order is =

**ab**'c" -**ab**"c'+a'b "c - a'bc" - a"bc' - a"b'c. 3.But his enemies increased in power, and about 1194 he was driven from Wales by the partisans of his half-brother Llewelyn

**ab**Iorwerth.Take a rod LMN bent at right angles at M, such that MN=

**AB**; let the leg LM always pass through a fixed point 0 on**AB**produced such that OA = CA, where C is the middle point of**AB**, and cause N to travel along the line perpendicular to**AB**at C; then the midpoint of MN traces the cissoid.The cartesian equation, when A is the origin and

**AB**= 2a, is y 2 (2a - x) =x 8; the polar equation is r= 2a sin 0 tan 0.The form is " A B C D, &c., are P;

**AB**CD are all M; thus all M are P."But that the origins of such mythical beings were,

**ab**initio, either solar or vegetable, or, for that matter, animal, it would often be hard to prove.He also wrote or edited various Chinese works on geography, the celestial and terrestrial spheres, geometry and arithmetic. And the detailed history of the mission was drawn out by him, which after his death was brought home by P. Nicolas Trigault, and published at Augsburg, and later in a complete form at Lyons under the name De Expeditione Christiana apud Sinas Suscepta,

**ab**Soc. Jesu, Ex P. Mat.A lowresistance galvanometer is connected by a very fine wire (2 to 3 mils) to the centre C of the experimental wire

**AB**, and also to the middle point D of the parallel wire so as to form a Wheatstone bridge.The balance is adjusted by shunting either AD or BD with a box, S, containing 20 to Ioo ohms. All the wires in the quadrilateral must be of the same metal as

**AB**, to avoid accidental thermoelectric effects which would obscure the result.If the current flows from A to B there will be heat absorbed in AC and evolved in CB by the Thomson effect, if the specific heat of electricity in

**AB**is positive as in copper.This is shown by the curve

**AB**, and is the useless work represented by the expression (q 2 -q 1)/r.The GAMESS-UK Benchmark is designed to represent the typical range of calculations commonly performed by the

**ab**initio quantum chemist.Most

**ab**initio groups quickly become very close-knit since students see each other almost every day.Unsure about my new contraceptive coil Will

**ab**toner affect the coil?They describe an English speaking

**ab**initio student of German who exhibited an unusually strong tendency to use word coinage and lexical innovation.Doing abdominal crunches on the

**AB**Revolutionizer is a perfect abdominal workout.The calories expended during

**ab**exercises are very low.Passed by voice not obligated to must be willing

**ab**to require.Still problems with pt retrieval Some success with constituent retrievals using

**AB**microwindows (not all MWs processed ).Patients were further subdivided into those with or without elevated quantitative serum Chlamydia trachomatis IgG antibody (

**Ab**) titres.Instant abs 6 second workout and lateral thigh stepper exercises, fitness &

**ab**lounge 2.. .Quintus Curtius was largely used for the Alexandreis (c. 1180) of Gaultier de Lille or de Chatillon (Galtherus

**ab**Insulis or de Castellione).Calling the foci S, S', the real vertices A, A', the extremities of the conjugate axis B, B' and the centre C, the positions of B, B' are given by

**AB**=**AB**' = CS.Ita Cornua Ammonis, quae ex nautilorum numero habeantur, passim et forma et magnitudine (nam et pedali diametro aliquando reperiuntur)

**ab**omnibus illis naturis discrepare dicunt, quas praebet mare.It was published (1589) by Giacomo Castelvetri, who had married his widow, with the title Explicatio gravissimae quaestionis utrum excommunicatio, quatenus religionem intelligentes et amplexantes, a sacramentorum usu, pro pier admissum facinus arcet, mandato nitatur divino, an excogitata sit

**ab**hominibus.This belief is, of course, not specifically Christian; it has been held at all times and everywhere by men of the most various races and creeds; and, if there be any validity in the contention that that is true which has been held semper, ubique, et

**ab**omnibus, no fact is better established.For the quadratic aoxi +2a i x i x 2 +a 2 x, we have (i.) ax = 7/1x1+2aixix2-I-7/24, (ii.) xx=xi+xzi (

**ab**) 2 =2(aoa2 - ai), a a = a o+712, _ (v.) (xa)ax= i'?- (a2 - ao)xix2 - aix2.The commutative law in arithmetic, for instance, states that adb and b+a, or

**ab**and ba, are equal.It had probably not been completed, though it was already defensible, when Prince Llewelyn

**ab**Griffith, incensed by its construction and claiming its site as his own, laid siege to it in 1271 and refused to retire except on conditions.If we bisect

**AB**and CD in P and Q respectively, and describe the arc PQ of a circle with centre 0, the length of this arc is 2(b+a)0; and b - a =**AB**.Then (see Parabola) Tc = Cv, Av = Vb, and K

**ab**is parallel to**AB**, so that aC = Cb.As

**AB**again moves from CD towards EF another compression or push is sent out, as it returns from EF towards CD another extension or pull, and so on.Thus waves are propagated along OX, each wave consisting of one push and one pull, one wave emanating from each complete vibration to and fro of the source

**AB**.The marriage was declared invalid

**ab**initio either on the ground of Anne's precontract with Lord Percy or more probably on the ground of the affinity established between Henry and Anne by Henry's previous relations with Mary Boleyn.Furthermore it is seen that

**AB**is perpendicular to the line joining the centres, and divides it in the ratio of the squares of the radii.Of games the young Moors play a great number; the principal one is a kind of football, more like that of Siam and Burma than that of England; wrestling and fencing are popular, but the chief amusement of the adult Moors is the "powder-play" (la `

**ab**el bariid), which consists of a type of military tournament, the horsemen going through lance and musket exercises or charging in review fashion, firing volleys as they gallop. Other recreations much in favour throughout Morocco are music, singing, jugglery, snake-charming and acrobatic performances.If we mark off a segment

**AB**along the line of action so as to represent the force completely, the A moment is represented as to magni FIG 15 tude by twice the area of the triangle OAB, and the usual convention as, .o sign is that the area is to be reckoned positive or negative iccording as the letters 0, A, B, occur in counter-clockwise or clockwise order.In 1613 he published his first printed work, though not his first literary composition - Gravissimae Quaestionis de Christianarum Ecclesiarum, in Occidentis praesertim partibus,

**ab**Apostolicis temporibus ad nostram usque aetatem, continua successione et statu, Historica Explicatio, wherein he took up the history of the Western Church from the point where Jewel had left off in his A pology for the Church of England, and carried it on from the 6th till past the middle of the 13th century, but never completed it.Still problems with pT retrieval Some success with constituent retrievals using

**AB**microwindows (not all MWs processed).Instant abs 6 second workout and lateral thigh stepper exercises, fitness &

**ab**lounge 2...If so, why not cover up that not quite a six pack

**ab**set you've got goin' on and give women a little something good to talk about?A person who inherits an A antigen gene from each parent has type A blood; receiving two B antigen genes corresponds with type B blood; and inheriting A and B antigen genes means a person has type

**AB**blood.Mainly, because people with Type

**AB**reportedly have low stomach acid, they are unable to efficiently break down animal products.With so many

**ab**workouts advertised on television, you hear about everything from "amazing" fad diets to the lady down the street who has her own miracle solution.You can build your

**ab**muscles up, but those efforts are hardly noticed if there's still a layer of fat covering those muscles.Strong

**ab**muscles support weightlifting by helping to keep your body stable.You can repeat these

**ab**exercises as many times as you like until you get tired.The video is fairly short and shows you some basic

**ab**exercises that you can perform at home.**Ab**workouts on an exercise ball are shown to be more effective than mat workouts.Having a toned and fit

**ab**section is possible for anyone by becoming familiar with 100 abdominal exercises.Don't worry, you don't have to perform every single one of these moves, but by incorporating a variety of

**ab**exercises into your routine, you can sculpt and tone your midsection just in time for swimsuit season.This way, you work your entire

**ab**section.Use your

**ab**muscles to perform the movement.Following a regular routine that features aerobic exercise a minimum of 30 minutes three times per week will help your

**ab**definition show.As with any weight-bearing exercises, don't perform these moves on consecutive days, but allow 48 hours to pass between

**ab**workouts.The most obvious and perhaps well-known

**ab**exercise is the crunch.Training the

**ab**muscles is greatly beneficial, not least for health reasons, but if your goal is visible washboard abs you're going to have to do something to burn off bodyfat too.When picking out an

**ab**exercise dvd, it's important to have your goals clear.That doesn't mean, however, that they will all have the same effect, and if you are taking the time to purchase an

**ab**exercise DVD, it is probably because you have a specific goal in mind.In fact, by targeting the abdominal muscles through an

**ab**exercise dvd it is entirely possible to make your stomach larger, because the muscles will be growing bigger as all muscles do through exercise.Lower

**ab**exercises are an important aspect of any complete exercise regime.Lower

**ab**exercises target muscles below the bellybutton.Luckily there are many lower

**ab**exercises to choose from, making it easier for people to switch up their routine periodically in an effort to continually challenge the lower abdominal muscles.Some of the best lower

**ab**exercises don't involve any equipment at all and can be done just about anywhere.Plenty of manufacturers have created equipment to accompany lower

**ab**exercises, but not all of them work equally well.Exercise balls are often utilized as part of a lower

**ab**routine.Exercise machines made specifically for abdominal work will include many different options for lower

**ab**exercises.Some personal trainers recommend that people with back problems concentrate on strengthening their body's core with lower

**ab**exercises in conjunction with other exercises designed to make the entire body stronger and less prone to injury.It is a good idea to speak to a personal trainer or physical therapist to find out what lower

**ab**exercises are appropriate for you with regards to your back problems.Don't just jump into a lower

**ab**routine that you find in a magazine or online, if you have existing back problems, because there is a chance that you can make your injuries worse instead of better.Considering an

**AB**Rocker for yourself or someone else?Based out of Greenwich, Connecticut, Adrian agreed to discuss the

**AB**Rocker with ''LoveToKnow'' Exercise.While I've never used this specific product, it seems to work the same way as most other sold-on-tv

**ab**products.By merely balancing on the Swiss ball, you are automatically engaging your core which is a better

**AB**workout for you before you even begin crunches and other**AB**exercises.It seems that the

**AB**Rocker product is similar to many of the other 'as seen on TV' exercise products that claim to tone the core.With the addition of an adjustable

**ab**bench or other simple weight bench, you can get a full workout for both your core and extremities by following the guidelines of either videos, books, or just a dumbbell exercise chart.The most common women's

**ab**exercises are different types of crunches and sit-ups.So even if losing enough fat to actually see those abs is out of the question,

**ab**exercises are still highly beneficial.Doing crunches and other

**ab**exercises ad nauseum will not magically melt off the fat covering them, revealing flat abs in the process.So, you're looking for a free

**ab**workout that will set your midsection on fire?For best results, switch things around a little from workout to workout, for example replacing The Plank with an

**ab**machine or doing The Bicycle instead of regular crunches.This is one of the more advanced components of your free

**ab**workout.For example, take a look at the free

**ab**workout offered by MK Danielsson.An effective, free lower

**ab**workout can be just what you need to tighten up that midsection for summer.Here are five exercises that make for a free lower

**ab**workout that will leave you good and sore for sure!**Ab**exercises may not be your favorite, however there are tons of free**ab**workout routine options because it's so easy to work the abdominals without expensive equipment.Your own body weight and effective use of your breath are more than sufficient to get a good

**ab**workout.The only way for you (and others) to see all the hard work you've been doing through a free

**ab**workout routine is to lose that extra layer of fat.An

**ab**workout routine can be as basic or as complicated as your fitness level permits.For this reason, the

**ab**exercises selected here are divided into a beginner/intermediate and an intermediate/advanced workout.In addition to these free options, it's also an idea to invest in an

**ab**exercise DVD if you find it difficult to envision how these exercises look in practice.There are literally hundreds of different

**ab**blasters out there, so any list of the best abdominal exercises around is bound to look a bit different depending on whom you ask.There's plenty more excellent

**ab**destroyers out there, such as rollerball, plank sidestands, Captain's Chair, dumbbell sidebends, reverse crunches, hanging knee raises and a myriad exercise ball-based ideas.In the eternal quest for the perfect, six-pack abs, many people consider buying

**ab**workout equipment.If you watch late-night television, you have probably seen numerous commercials and infomercials for

**ab**workout equipment.The chair or lounge type of

**ab**machine is designed to solve the problems associated with traditional abdominal exercises.Although these

**ab**exercise machines have different names, they all work in a similar manner.The Six Second

**Ab**machine has a clicking device, which provides three clicks for the flexing phase of the crunch and three clicks for the return.Examples include the

**Ab**Slide, the**Ab**Dolly and the Torso Track.The American Council on Exercise, otherwise known as ACE, was also curious about the effectiveness of popular

**ab**exercise machines.They were asked to perform a variety of abdominal exercises on various types of

**ab**equipment.These included the Torso Track, the**Ab**Rocker and the**Ab**Roller.Additionally, the

**Ab**Roller scored the same as the traditional crunch and the**Ab**Rocker was 80 percent less effective than the crunch.The bicycle maneuver, which is performed without

**ab**exercise equipment was one of the most effective effective exercises for both the rectus abdominus and the obliques.The stability ball, which usually costs under $30 was one of the most effective pieces of

**ab**workout equipment.Consider these findings before you invest over $200 on an

**ab**machine.The

**Ab**Tronic,**AB**Energizer and Fast Abs were the Commission's first targets.The home exercise machines included the Torso Track, the

**Ab**Roller and the**Ab**Rocker.The

**Ab**Roller claimed ninth place for the rectus abdominus and tenth place for the obliques.The

**ab**roller was in 13th or last place for both muscle groups.As such, a Bowflex, a Total Gym or adjustable dumbbells will provide better value than an

**Ab**Lounge.When working out your abdominal muscles, performing lower

**ab**crunches can help you to avoid neglecting this all-important part of your core muscle group.Lower

**ab**crunches are an essential part of exercising the muscles of your core.Performing lower

**ab**crunches strengthens the lower part of your rectus abdominis and your transverse abdominis.For an added

**ab**workout, do your dumbbell routine using an exercise or stability ball.Those looking for sleek, built abs should consider the

**ab**lounge sport with its "work smarter, not harder" mantra.When body fat is low and these muscles are highly defined through targeted exercise like that of the

**ab**lounge sport variety, the result is what many people call "six-pack abs."This

**ab**machine allows you to get in a targeted**ab**workout from the comfort of your home.The

**ab**exerciser allows you to work both upper and lower abs.**Ab**machines are not quick fixes and simply offer an**ab**workout alternative to traditional floor**ab**exercises.This particular

**ab**machine comes with a workout DVD, as well as written instructions that can give you an idea of where to begin with your exercise regimen.It's a good idea to consult a doctor or trainer if you haven't exercised recently to see if an

**ab**machine workout is right for you.There are many excellent

**ab**workouts for men.Though crunches are very popular, there are a number of

**ab**workouts for men that will not only work the core area, but may also help to tone other parts of the body, as well.With only a little time, and a floor mat, it's possible to get great lower

**ab**workouts at home.In truth, working your lower abs is going to help pull your stomach in a lot faster than doing standard crunches or

**ab**exercises alone.Simply incorporating some lower

**ab**exercises into your weekly routine can help you get the results you're after in a much shorter span of time.This muscle group is what pulls your belly inward; targeting this lower

**ab**group is what's going to help flatten your stomach.You don't need the equipment at the gym to get a great lower

**ab**workout.Great lower

**ab**workouts at home can help you achieve the results you're after in less time than it takes to get to the gym.If you want to tone your tummy while improving balance and stability, subscribing to a program of easy

**ab**workouts may be all you need.Remember to concentrate on form and consistency when performing easy

**ab**workouts.Add an

**ab**routine to your program every other day for roughly five to ten minutes.The 6 best

**ab**exercises come from a variety of exercise styles, giving you several options for working your core to the max.Of the 6 best

**ab**exercises, you may find this one the most challenging.Yoga offers several ways to work your

**ab**muscles which are non-jarring and controlled.For a more advanced workout to complete the 6 best

**ab**exercises, you can try the ball tuck-in exercise, sometimes also called the jackknife.Any abdominal exercises will help you to strengthen your

**ab**region.Improperly performing

**ab**exercises could lead to back pain or injury; plus, you'll actually see greater results when you perform the exercises with the correct form.The eyepiece

**ab**consists of two plano-convex lenses a, b, of nearly the same focal length, and with the two convex sides facing each other.Now, if CD is pressed by its weight or by a spring on the surface

**AB**, the effect of wear will be to produce a symmetrical grinding away of both surfaces, which may be represented thus, fig.Some see the guarantee, or at least the indication, of infallibility in the consensus of the Church (quod semper, ubique, et

**ab**omnibus) expressed from time to time in general councils; others see it in the special grace conferred upon St Peter and his successors, the bishops of Rome, as heads of the Church; others again see it in the inspired Scriptures, God's Word.It was destroyed in 1260 by Llewellyn

**ab**Gruffydd, prince of Wales, with the supposed connivance df Mortimer, but its site was reoccupied by the earl of Lincoln in 277, and a new castle at once erected.Leading examples may be found in the various prosecutions of St Athanasius, in whose case also there is the germ of an appeal, tanquam

**ab**abusu.The appellatio tanquam

**ab**abusu (appel comme d'abus) in France was an application of a like nature.His father belonged to the class of Dihkans (the old native country families and landed proprietors of Persia, who had preserved their influence and status under the A

**ab**rule), and possessed an estate in the neighbourhood of Cus (in Khorasan).Among these posts were various procuratorships (chief of which was that of the imperial fisc), and the offices

**ab**epistulis, a rationibus and a libellis (secretary, accountant, receiver of petitions).Let the curve represent an elliptic orbit,

**AB**being the major axis, DE the minor axis, and F the focus in which the centre of attraction is situated, which centre we shall call the sun.These early fortifications of the Acropolis, ascribed to the primitive non-hellenic Pelasgi, must be distinguished from the Pelasgicum or Pelargicum, which was in all prob

**ab**i l i ty an encircling wall, built round the base of the g citadel and furnished with nine gates from which it derived the name of Enneapylon.In it he discusses the "notes" which distinguish Catholic truth from heresy, and (cap. 2) lays down and applies the famous threefold test of orthodoxy - quod ubique, quod semper, quod

**ab**omnibus credi-tum est.If two solutions containing the salts

**AB**and CD be mixed, double decomposition is found to occur, the salts AD and CB being formed till a certain part of the first pair of substances is transformed into an equivalent amount of the second pair.The proportions between the four salts

**AB**, CD, AD and CB, which exist finally in solution, are found to be the same whether we begin with the pair**AB**and CD or with the pair AD and CB.Let the determinant of the b's be

**Ab**and B rs, the minor corresponding to b rs .It should be noticed that the real function denoted by (

**ab**) 2 is not the square of a real function denoted by (**ab**).For a single quantic of the first order (

**ab**) is the symbol of a function of the coefficients which vanishes identically; thus (**ab**) =a1b2-a2bl= aw l -a1ao=0 and, indeed, from a remark made above we see that (**ab**) remains unchanged by interchange of a and b; but (**ab**), = -(ba), and these two facts necessitate (**ab**) = o.In either case (

**AB**) =A 1 B 2 -A 2 B 1 = (A/2)(**ab**); and, from the definition, (**ab**) possesses the invariant property.Since (

**ab**) = a l b 2 -a 2 b l, that this may be the case each form must be linear; and if the forms be different (**ab**) is an invariant (simultaneous) of the two forms, its real expression being aob l -a l b 0.When either of the forms is of an order higher than the first (

**ab**), as not being expressible in terms of the actual coefficients of the forms, is not an invariant and has no significance.The expression (

**ab**) 4 properly appertains to a quartic; for a quadratic it may also be written (**ab**) 2 (cd) 2, and would denote the square of the discriminant to a factor pres.For the quartic (

**ab**) 4 = (aib2-a2b,) alb2 -4a7a2blb2+64a2 bib2 - 4a 1 a 2 b 7 b 2 + a a b i = a,a 4 - 4ca,a 3 +6a2 - 4a3a3+ aoa4 = 2(a 0 a 4 - 4a1a3 +e3a2), one of the well-known invariants of the quartic.I begin

**ab**ovo.