B Sentence Examples

Part B outlines the new policy, which promotes sustainable agriculture, and Part C focuses on policy implementation.

If taken with other metabolites that aid in the creation of serotonin - vitamin B, niacin and magnesium, which a multivitamin should provide - 5HTP can help deal with many of the same health issues that people take tryptophan for.

In the apparatus of type B as made by Zeiss there are two microscopes attached to a base-plate, one of which views the spectrum-plate (or other object) to be measured, while the other views a scale that moves with the slide on which the spectrumplate is mounted.

But the day showed the ' B attle of Macedonian army equal to the task.

Many organic compounds of boron are known; thus, from the action of the trichloride on ethyl alcohol or on methyl alcohol, ethyl borate B(OC2H5)3 and methyl borate B(OCH 3) 3 are obtained.

The next stage (b) is connected with the suppression of the local high-places or minor shrines in favour of a central sanctuary.

Nutritional supplements that may be beneficial include large amounts of antioxidants (vitamins C, A, E, zinc, selenium, and flavenoids), as well as B vitamins and a full complement of minerals (including boron, copper, and manganese).

Any note may be a pitch note; for orchestras custom has settled upon a' in the treble clef, for organs and pianos in Great Britain c 2, and for modern brass instruments b flat'.

The exact meaning of these features is not clear, but if it be remembered (a) that the Levites of post-exilic literature represent only the result of a long and intricate development, (b) that the name "Levite," in the later stages at least, was extended to include all priestly servants, and (c) that the priesthoods, in tending to become hereditary, included priests who were Levites by adoption and not by descent, it will be recognized that the examination of the evidence for the earlier stages cannot confine itself to those narratives where the specific term alone occurs.

Although married she always remained a member of her father's house - she is rarely named wife of A, usually daughter of B, or mother of C. Divorce was optional with the man, but he had to restore the dowry and, if the wife had borne him children, she had the custody of them.

At small country towns or villages, where the message traffic is light, the Wheatstone " A B C " instrument is used.

In this apparatus electric A B C currents are generated by turning a handle (placed in front of the instrument), which is geared, in the instru ment.

The disaster which aroused Nehemiah's grief was scarcely the fall of Jerusalem in 586 B.C., but a more recent one, and it has been conjectured that it followed the work of Ezra (in b above).

To start with the last before considering (a) and (b) would be futile.

Thus, in any estimate of the influence of Babylonia upon the Old Testament, it is obviously necessary to ask whether certain features (a) are of true Babylonian origin, or (b) merely find parallels or analogies in its stores of literature; whether the indebtedness goes back to very early times or to the age of the Assyrian domination or to the exiles who now returned.

The later class (B) of the linear script is that used on the great bulk of the clay tablets of the Cnossian palace, amounting in number to nearly 2000.

His extant works are (a) three poems, "The Praises of Wemen" (224 lines), "On Luve" (10 lines), and "The Miseries of a Pure Scholar" (189 lines), and (b) a Latin account of the Arbuthnot family, Originis et Incrementi Arbuthnoticae Familiae Descriptio Historica (still in MS.), of which an English continuation, by the father of Dr John Arbuthnot, is preserved in the Advocates' Library, Edinburgh.

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

Thus " the son of the king " is more commonly expressed by b`ra dh`malka or b`reh d`malka than by bar mailed, whereas the latter type would alone be permissible in Hebrew.

The conformation of those flowers a consists essentially in the pres- ' 'A B ence of a six-parted perianth, the three outer segments of which correspond to a calyx, the three inner ones to a corolla.

E Arabian Sea Ba Of G A L e Geological information incomplete Desert Deposits Quaternary Tertiary Mesozoic Palaeozoic Archaean and Metamorphic Younger Volcanic Rocks English Miles b iuHi iiiiuiiiiii after llargl,aua Geology The geology of Asia is so complex and over wide areas so little known that it is difficult to give a connected account of either the structure or the development of the continent, and only the broader features can be dealt with here.

The prostomium overhangs the mouth, and is often of considerable size and, as a rule, quite distinct from the segment following, being A„ f s / 6/ ' A B I.

B, Diagram of the nephridium of Alciope, into which opens the large genital funnel (coelo mostome).

Here, however, the buds are lateral, though produced from a budding may be defective upon one or other of the noto a b FIG.

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.

In the case of plants the method of procedure was to grow some of the most important crops of rotation, each separately year after year, for many years in succession on the same land, (a) without manure, (b) with farmyard manure and (c) with a great variety of chemical manures; the same description of manure being, as a rule, applied year after year on the same plot.

Amongst the field experiments there is, perhaps, not one of more universal interest than that in which wheat was grown for fifty-seven years in succession, (a) without manure, (b) with farmyard manure and (c) with various artificial manures.

In 1890, at Plymouth, competitions took place of light portable engines (a) using solid fuel, (b) using liquid or gaseous fuel, grist mills for use on a farm, disintegrators, and cider-making plant for use on a farm.

In 1892, at Warwick, the competitions related to ploughs - single furrow (a) for light land, (b) for strong land, (c) for press drill and broad-cast sowing; two-furrow; three-furrow; digging (a) for light land, (b) for heavy land; and one-way ploughs.

In 1899, at Maidstone, special prizes were offered for machines for washing hops with liquid insecticides, cream separators (power and hand), machines for the evaporation of fruit and vegetables, and packages for the carriage of (a) soft fruit, (b) hard fruit.

A B In Great Britain the flea beetles (Halticidae) are one of the most serious enemies; one of these, the turnip flea (Phyllotreta nemorum), has in some years, notably 1881, caused more than 500,00o loss in England and Scotland alone by eating the young seedling turnips, cabbage and other Cruciferae.

These parasites damage the hide, B FIG.

A, Winged female; B, winged D, viviparous wingless female from in patches from old apple trees, where the insects live in the rough bark and form cankered growths both above and below ground.

A, branch bearing male cones, reduced; B, single male cone, enlarged; C, single stamen, enlarged.

A variety of the spruce, abounding in some parts of Nor way, produces a red heartwood, not easy to distinguish from that of the Norway B pine (Scotch fir), and imported with it into England as "red deal" or "pine."

A, Branch bearing (a) young female cones, (b) ripe cones, reduced.

B, Ripe cone scale with seeds, enlarged.

In the subsequent growth of ergc- pl.y- i ln b a.

Neither the rotation of the shell as a whole nor its helicoid spiral coiling is the immediate cause of the torsion of the body in the individual, for the direction of the torsion is indicated in the segmentation of the ovum, in which there is a complete A B From Lankester's Treatise on Zoology.

B, The same, partially everted by eversion of the sides, as in the Nemertine proboscis and Gastropod eye-tentacle = pleurecbolic.

The introvert is not a simple one with complete range both in eversion and introversion, but is arrested in introversion by the fibrous bands at c, and similarly in eversion by the fibrous bands at b.

G, The acrecbolic snout of a proboscidiferous Gastropod, arrested short of complete eversion by the fibrous band b.

On the other hand, many which have a siphonate mantle-skirt are not provided with an introversible mouth-bearing al- B c..

B, Sole of the foot of Pyrula tuba, to show a, the pore usually said to be " aquiferous " but probably the orifice of a gland; b, median line of foot.

B, The diblastula has become a trochosphere by the development of the ciliated ring y r (optical section).

B, The shell removed.

Posterior cephalic tentacle; and b, the eyes.

Ia, frons; b, clypeus (the pointed labrum beneath it); II, mandible; III, first maxilla; (a, base; b, sheath; c, piercer), III', inner view of sheath; IV, second maxillae forming rostrum (b, mentum; c, ligula).

B, Section through compound eye (after Miall and Denny); C, organs of smell in cockchafer; (after Kraepelin); D, a, b, sensory pits on cercopods of golden-eye fly; c, sensory pit on palp of stone-fly (after Packard); E, sensory hair (after Miall and Denny); F, ear of long-horned grasshopper; a, Front shin showing outer opening and air-tube; b, section (after Graber); G, ear of locust from within (after Graber).

To some of the Linnaean genera he dare not, however, assign a place, for instance, Buceros, Haematopus, Merops, Glareola (B risson's genus, by the by) and Palamedea.

Under the caliphate of Mamun, Saman, a Persian noble of Balkh, who was a close friend of the Arab governor of Khorasan, Asad b.

His son Asad, named after Asad b.

A=D= -_-- - - ---Island =r= b = o =ir- monument by James Edward Kelly to General Fitz John Porter; a cottage hospital (1886); a United States naval hospital (1891); a home for aged and indigent women (1877); and the Chase home for children (1877).

If a be greater than b the curve resembles fig.

This letter corresponds to the second symbol in the Phoenician alphabet, and appears in the same position in all the European alphabets, except those derived, like the Russian, from medieval Greek, in which the pronunciation of this symbol had changed from b to v.

A new form had therefore to be invented for the genuine b in Slavonic, to which there was, at the period when the alphabet was adopted, no corresponding sound in Greek.

The new symbol, which occupies the second position, was made by removing the upper loop of B, thus producing a symbol somewhat resembling an ordinary lowercase b.

The old B retained the numerical value of the Greek i as 2, and no numerical value was given to the new symbol.

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

In Late Latin there was a tendency to this spirant pronunciation which appears as early as the beginning of the 2nd century A.D.; by the 3rd century b and consonantal u are inextricably confused.

He may lease the settled land, or any part of it, for any time not exceeding (a) in the case of a building lease, 99 years; (b) in the case of a mining lease, 60 years, (c) in the case of any other lease, 21 years.

This enactment applies to leases of agricultural subjects, houses, mills, fisheries and whatever is fundo annexum; provided that (a) the lease, when for more than one year, must be in writing, (b) it must be definite as to subject, rent (which may consist of money, grain or services, if the reddendum is not illusory) and term of duration, (c) possession must follow on the lease.

If at least half of the harvest in any year is destroyed by accident, the lessee (a) in the case of a lease for several years, obtains, at the end of his lease, a refund of rent, by way of indemnity, unless he has been indemnified by preceding harvests; (b) in the case of a lease for a year only, may secure a proportional abatement of the current rent.

Branches were promoted (a) from Mukden to Antung on the Yalu, to connect with the Korean system, and (b) from Kwang-cheng-tsze to Kirin.

To it belong (a) superficial grooves or deeper slits situated on the integument near the tip of the head, (b) nerve lobes in immediate connexion with the nervous tissue of the brain, and (c) ciliated ducts penetrating into the latter and communicating with the former.

Two pairs of invaginations of B the skin, which originally are called the prostomial and metastomial disks, grow round the intestine, finally fuse together, and form the skin and mus- cular body-wall of the future Nemertine, which afterwards becomes ciliated, frees itself from the pilidium investment and develops into the adult worm without further metamorphosis.

If these conditions are not fulfilled sufficiently, the ammeter will not give the same indications for the same current if that current has been reached (a) by increasing from a smaller current, or (b) by decreasing from a larger current.

Let -I- B, - B, be two smaller trunnions which project out from the sides of the two strips connecting together a pair of rings CC. The rings and the connecting strips constitute the circuit which is to be rendered movable.

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

The processes of soap manufacture may be classified (a) according to the temperatures employed into (I) cold processes and (2) boiling processes, or (b) according to the nature of the starting material - acid or oil and fat - and the relative amount of alkali, into (1) direct saturation of the fatty acid with alkali, (2) treating the fat with a definite amount of alkali with no removal of unused lye, (3) treating the fat with an indefinite amount of alkali, also with no separation of unused lye, (4) treating the fat with an indefinite amount of alkali with separation of waste lye.

From the properties of the ellipse, A is the pericentre or nearest point of the orbit to the centre of attraction and B the apocentre or most distant point.

Since the distance of a body from the observer cannot be observed directly, but only the right ascension and declination, calling these a and 6 we conceive ideal equations of the form a = f (a, b, c, e, f, g, t) and 5=0 (a, b, c, e, f, g, t), the symbols a, b,.

The oldest mention of Robin Hood at present known occurs in the second edition - what is called the B text - of Piers the Plowman, the date of which is about 1377.

Henry started with methyl iodide, the formula of which we write in the form CI a H b H c H d.

The same methyl iodide gave with potassium cyanide, acetonitril, which was hydrolysed to acetic acid; this must be C(Coch) a H b H c H d.

This acid with silver nitrite gave nitroacetic acid, which readily gave the second nitromethane, CH a (NO 2) b H c H d, identical with the first nitromethane.

B aeyer has suggested that his hypothesis may also be applied to explain the instability of acetylene and its derivatives, and the still greater instability of the polyacetylene compounds.

The space a must allow for the inclusion of a copper spiral if the substance contains nitrogen, and a silver spiral if halogens be present, for otherwise nitrogen oxides and the halogens may be condensed in the absorption apparatus; b contains copper oxide; c is a space for the insertion of a porcelain or platinum boat containing a weighed quantity of the substance; d is a copper spiral.

The magnesite (a) serves for the generation of carbon dioxide which clears the tube of air before the compound (mixed with fine copper oxide (b)) is burned, and afterwards sweeps the liberated nitrogen into the receiving vessel (e), which contains a strong potash solution; c is coarse copper oxide; and d a reduced copper gauze spiral, heated in order to decompose any nitrogen oxides.

Eliminating a and b between these relations, we derive P k V k /Tk= 8R, a relation which should hold between the critical constants of any substance.

Van der Waal's equation (p-I- a/v 2) (v - b) = RT contains two constants a and b determined by each particular substance.

To reduce these figures to a common standard, so that the volumes shall contain equal numbers of molecules, the notion of molecular volumes is introduced, the arbitrary values of the crystallographic axes (a, b, c) being replaced by the topic parameters' (x, ?i, w), which are such that, combined with the axial angles, they enclose volumes which contain equal numbers of molecules.

From B the curve of equilibrium (BD) between rhombic and liquid sulphur proceeds; and from C (along CE) the curve of equilibrium between liquid sulphur and sulphur vapour.

He claimed descent from the caliph Abubekr, and from the Khwarizm-Shah Sultan `Ala-uddin b.

B, The proboscis of one still more highly magnified.

B, A section through the same; a, the invaginated proboscis; b, proboscis sheath; c, beginning of the neck; d, lemniscus.

Dulceti, 1339, and b, On Mercator's projection, according to modern maps.

The Somali, however, declare themselves to be of Arab origin, alleging their progenitor to have been a certain Sherif Ishak b.

Climate (B) is the steady winter climate of Edmonton district.

The earliest literary papyrus is that known, from the name of its former owner, as the Prisse papyrus, and now preserved at Paris, containing a work composed in the reign of a king of the fifth dynasty, and computed to be itself of the age of upwards of 2500 years B,C. The papyri discovered in Egypt have often been found in tombs, and in the hands, or swathed with the bodies, of mummies.

The cheek-teeth (premolars and molars) form a A B C FIG.

The Zohar pretends to be a compilation made by Simon b.

If a and b are cardinal numbers, and a+I =b+1, then a =b.

For superior education there is (1) the university of Constantinople, with its four faculties of letters, science, law and medicine; and (2) special schools, including (a) the normal school for training teachers, (b) the civil imperial school, (c) the school of the fine arts and (d) the imperial schools of medicine.

The artillery is divided into (a) field artillery, horse artillery, mountain artillery and howitzer regiments; (b) fortress artillery; (c) artillery depots.

In this manner an annuity of £T159,500 was set free, of which £Ti i,000 per annum was allotted as " extraordinary sinking fund " to series A and £T49,500 per annum each to series B, C and D; the lottery bonds were originally excluded from this arrangement, and special compensation was granted to these later.

Series B, C and D (series A having already been completely redeemed by the action of the sinking fund) were replaced by the creation of new 4% bonds to a nominal amount of £T32,738,772, with a sinking fund of 0.45% per annum, bearing identical rights and privileges, and ranking immediately after, the priority bonds.

The rates at which the series were respectively exchanged against the new unified bonds were £loo series B against £70 unified, £loo series C against £42 unified and £too series C against £37, 10s.

Table B shows the total indebtedness of the Ottoman Empire, exclusive of tribute loans.

Dresden was the last great victory of the First Empire, By noon on the 27th August the Austrians and Russians were completely beaten and in full retreat, the French pressing hard behind them, but meanwhile Napoleon himself again succumbed G Beereri B eip \ ii g?

On the i i th and 12th of Battles December there were engagements of a less severe before character, and finally on the 13th of December Soult B, or Battles ayonne of with 35,000 men made a vehement attack up the the Nive, right bank of the Nive against Hill, who with about Dec. 10-13, 14,000 men occupied some heights from Villefranque 1813.

In 734 their king Sanip(b)u was a vassal of Tiglathpileser IV., and his successor, P(b)udu-ilu, held the same position under Sennacherib and Esarhaddon.

Under the influence of an applied electric force, he imagined that the B part of the first molecule was liberated at the anode, and that the A part thus isolated united with the B part of the second molecule, which, in its turn, passed on its A to the B of the third molecule.

In this manner, the B part of the last molecule of the chain was seized by the A of the last molecule but one, and the A part of the last molecule liberated at the surface of the cathode.

It will deal briefly (I) with the general idea and the historical evolution of ecclesiastical vestments, (2) with the vestments as at present worn (a) in the Roman Catholic Church, (b) in the Oriental Churches, (c) in the Reformed Churches, (d) in the Anglican Church.

They comprise (a) the older works surrounding the town; (b) a group of coast batteries on the high ground of Cape Bizerta or Guardia, 4 m.

D, Lunaria biennis, showing the septum B, Lepidium sativum.

Engler's Versuch einer Entwickelungsgeschichte der Pflanzenwelt (Leipzig, 1879-1882), we should have in Siberia (a) the arctic region; (b) the sub-arctic or coniferous region - north Siberian province; (c) the Central-Asian domain - Altai and Daurian mountainous regions; and (d) the east Chinese, intruding into the basin of the Amur.

Pro-deltidium attached A B C FIG.

A, Larva which has just left brood-pouch; B, longitudinal section through a somewhat later stage; C, the fully formed embryo just before fixing - the neo-embryo of Beecher.

B, C, D, Stages showing the turning forward of the second or mantle segment.

The declaration is to the effect that the clergyman has not received the presentation in consideration of any sum of money, reward, gift, profit or benefit directly or indirectly given or promised by him or any one for him to any one; that he has not made any promise of resignation other than that allowed by the Clerical Resignation Bonds Act 1828; that he has not for any money or benefit procured the avoidance of the benefice; and that he has not been party to any agreement invalidated by sec. 3 sub-sec. 3 of the act which invalidates any agreement for the exercise of a right of patronage in favour or on the nomination of any particular person, and any agreement on the transfer of a right of patronage (a) for the retransfer of the right, or (b) for postponing payment of any part of the consideration for the transfer until a vacancy or for more than three months, or (c) for payment of interest until a vacancy or for more than three months, or (d) for any payment in respect of the date at which a vacancy occurs, or (e) for the resignation of a benefice in favour of any person.

In particular the square of a determinant is a deter minant of the same order (b 11 b 22 b 33 ...b nn) such that bik = b ki; it is for this reason termed symmetrical.

In the theory of surfaces we transform from one set of three rectangular axes to another by the substitutions 'X=' by+ cz, Y = a'x + b'y + c'z, Z =a"x+b"y-l-c"z, where X 2+Y2+Z2 = x2+ y2+z2.

There are then 2n(n-I) quantities b rs .

Let the determinant of the b's be Ab and B rs, the minor corresponding to b rs .

We may therefore form an orthogonal transformation in association with every skew determinant which has its leading diagonal elements unity, for the Zn(n-I) quantities b are clearly arbitrary.

Assuming then 01 to have the coefficients B1, B2,...B,, and f l the coefficients A 1, A21...A,n, we may equate coefficients of like powers of x in the identity, and obtain m+n homogeneous linear equations satisfied by the m+n quantities B1, 2, ...B n, A 1, A 2, ...A m.

Further, let 1 -1-b i x+ b 2 x 2' +...

Again, if a i, a 2, a 3 ...a m, be the t " roots of -1, b 1 = b 2 =...

If m be infinite and 1 + b i x + b 2 x 2 +...

We write;L 22 = a 1 a 2 .b 1 n-2 b2s 3 n - 3 3 n-3 3 n-3 3 a 3 = a 1 a 2 .b 1 b 2 .c 1 c2, and so on whenever we require to represent a product of real coefficients symbolically; we then have a one-to-one correspondence between the products of real coefficients and their symbolic forms. If we have a function of degree s in the coefficients, we may select any s sets of umbrae for use, and having made a selection we may when only one quantic is under consideration at any time permute the sets of umbrae in any manner without altering the real significance of the symbolism.

Such 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.

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 other words, B, b may be regarded as different or alternative symbols to A, a.

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.

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.

In general it will be simultaneous covariant of the different forms n 1 rz 2 n3 a, b x, ?

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.

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.

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.

In particular, when the product denotes an invariant we may transform each of the symbols a, b,...to x in succession, and take the sum of the resultant products; we thus obtain a covariant which is called the first evectant of the original invariant.

The first transvectant, (f,f') 1 = (ab) a x b x, vanishes identically.

Further, in the case of invariants, we write A= (1, i') 2 and take three new forms B = (i, T) 2; C = (r, r`) 2; R = (/y).

The discriminant is the resultant of ax and ax and of degree 8 in the coefficients; since it is a rational and integral function of the fundamental invariants it is expressible as a linear function of A 2 and B; it is independent of C, and is therefore unaltered when C vanishes; we may therefore take f in the canonical form 6R 4 f = BS5+5BS4p-4A2p5.

Ternary and Higher Forms.-The ternary form of order n is represented symbolically by (aixl+a2x2+a3x3)' =a'; and, as usual, b, c, d,...

We can see that (abc)a x b x c x is not a covariant, because it vanishes identically, the interchange of a and b changing its sign instead of leaving it unchanged; but (abc) 2 is an invariant.

When R =0, and neither of the expressions AC - B 2, 2AB -3C vanishes, the covariant a x is a linear factor of f; but, when R =AC - B 2 = 2AB -3C =0, a x also vanishes, and then f is the product of the form jx and of the Hessian of jx.

When a z and the invariants B and C all vanish, either A or j must vanish; in the former case j is a perfect cube, its Hessian vanishing, and further f contains j as a factor; in the latter case, if p x, ax be the linear factors of i, f can be expressed as (pa) 5 f =cip2+c2ay; if both A and j vanish i also vanishes identically, and so also does f.

Remark, too, that we are in association with non-unitary symmetric functions of two systems of quantities which will be denoted by partitions in brackets ()a, ()b respectively.

Solving the equation by the Ordinary Theory Of Linear Partial Differential Equations, We Obtain P Q 1 Independent Solutions, Of Which P Appertain To S2Au = 0, Q To 12 B U =0; The Remaining One Is Ab =Aobl A 1 Bo, The Leading Coefficient Of The Jacobian Of The Two Forms. This Constitutes An Algebraically Complete System, And, In Terms Of Its Members, All Seminvariants Can Be Rationally Expressed.

It Is Important To Notice That The Expression (0) A (0'Ls) B (01)A(0'18 1)B (812)A (0'18 2)B (Op).

T Aw 6(23)B A Series Of 2(W 2) Or Of 2(W I) Forms According As W Is Even Or Uneven.

For w = i the form is A i ai+Bib i, which we may write aob l -albo = ao(I) b -(I)abo; the remaining perpetuants, enumerated by z I - 2' have been set forth above.

For the case 8=1, 0' =2, the condition is a i r 1 72 = A032=0; and the simplest perpetuant, derived directly from the product A 1 B 21 is (I)a(2)b-(21)b; the remainder of those enumerated by z3 I z.

To represent the simplest perpetuant, of weight 7, we may take as base either A2B 1 B 2 or A l A 2 B2, and since Ai+Bi =o the former is equivalent to A 2 ArB 2 and the latter to A 2 B i B2; so that we have, (1 -f-aix) (1 + a2x).

By the rules adopted we take A?B 2 B 3, which gives (12)a(32)b - (1)a(321)b+ao(3212)b, the simplest perpetuant of weight 7; and thence the general form enumerated by the generating function 1 -z.1-z2.1 - z3 ?

The calculation results in -A,113 B2Bi+2A2B3B2Bi-AzB3BaBi+A4B3B1-2AlB3B2B1 -MB2B2131+MB33231+A213MB1 + A 2 B a Bi -2A2B3B2B +A2B1B1=0.

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.

The borate, Pb 2 B 6 0 1 u 4H20, is obtained as a white precipitate by adding borax to a lead salt; this on heating with strong ammonia gives PbB2044H2.

The most famous of these are B euves de Comarchis, Ernaud de Girone, Garin d'Anseun, Aimer le chaff, so called from his long captivity with the Saracens.

When the direction of any vector quantity denoted by a symbol is to be attended to, it is usual to employ for the symbol either a block letter, as H, I, B, or a German capital, as j,, 3?

If V denote the potential, F the resultant force, X, Y, Z, its components parallel to the co-ordinate axes and n the line along which the force is directed, then - sn = F, b?= X, - Sy = Y, -s Surfaces for which the potential is constant are called equipotential surfaces.

These two positions are sometimes called the first and second (or A and B) principal positions of Gauss.

Hence if the induction per square centimetre at any point is denoted by B, then in empty space B is numerically equal to H; moreover in isotropic media both have the same direction, and for these reasons it is often said that in empty space (and practically in air and other nonmagnetic substances) B and H are identical.

The magnetic flux per square centimetre at any point (B, B, or 0) is briefly called the induction, or, especially by electrical engineers, the flux-density.

The induction may therefore be specified as B lines per square centimetre.

I = (B - H)/47r, (24) or B = H +41rI.

Hence B =µH and I =KH.

Moreover, it is not constant, being an apparently arbitrary function of H or of B; in the same specimen its value may, under different conditions, vary from less than 2 to upwards of 5000.

The total magnetic induction or flux corresponds to the current of electricity (practically measured in amperes); the induction or flux density B to the density of the current (number of amperes to the square centimetre of section); the magnetic permeability to the specific electric conductivity; and the line integral of the magnetic force, sometimes called the magnetomotive force, to the electro-motive force in the circuit.

The angle B is indicated by the position of the spot of light upon the scale, and the horizontal intensity of the earth's field H E is known; thus we can at once determine the value of H P, from which the magnetization I of the body under test may be calculated.

The inner coil is supplied, through the intervening apparatus, with current from the battery of secondary cells B,; this produces the desired magnetic field inside the tube.

Since the induction B is equal to H 47rI, it is easy from the results of experiments such as that just described to deduce the relation between B and H; a curve indicating such relation is called a curve of induction.

The current passes through the rocking key K, which, when thrown over to the right, places a in contact with c and b with d, and when thrown over to the left, places a in contact with e and b with f.

On the other hand, the form of the third curve, with its large intercepts on the axes of H and B, denotes that the specimen to which it relates possesses both retentiveness and coercive force in a high degree; such a metal would be chosen for making good permanent magnets.

Ii shows the relation of B to H in a specimen which has never before been magnetized.

Denoting by W the work in ergs done upon a cubic centimetre of the metal (=_fHdB or f HdI), he finds W =nips approximately, where n 47r is a number, called the hysteretic constant, depending upon the metal, and B is the maximum induction.

Steinmetz's formula may be tested by taking a series of hysteresis curves between different limits of B,' measuring their areas by a pianimeter, and plotting the logarithms of these divided by 47r as ordinates against logarithms of the corresponding maximum values of B as abscissae.

But though a formula of this type has no physical significance, and cannot be accepted as an equation to the actual curve of W and B, it is, nevertheless, the case that by making the index e =1.6, and assigning a suitable value to r t, a formula may be obtained giving an approximation to the truth which is sufficiently close for the ordinary purposes of electrical engineers, especially when the limiting value of B is neither very great nor very small.

After pointing out that, since the magnetization of the metal is the quantity really concerned, W is more appropriately expressed in terms of I, the magnetic moment per unit of volume, than of B, he suggests an experiment to determine whether the mechanical work required to effect the complete magnetic reversal i Phil.

Baily showed that hysteresis ceased to increase when B was carried beyond 23,000.

This value of B corresponds to I =1640, the saturation point for soft iron.

Shimizu 3 indicate that Steinmetz's formula holds for nickel and annealed cobalt up to B =3000, for cast cobalt and tungsten steel up to B =8000, and for Swedish iron up to B =18,000, the range being in all cases extended at the temperature of liquid air.

Curves of magnetization (which express the relation of I to H) have a close resemblance to those of induction; and, indeed, since B = H+47r1, and 47rI (except in extreme fields) greatly exceeds H in numerical value, we may generally, without serious error, put I = B /47r, and transform curves of induction into curves of magnetization by merely altering the scale to which the ordinates are referred.

The loss for any induction B within the range for which Steinmetz's law holds may be converted into that for the standard induction 2500 by dividing it by B 6 /2500'.

If H l and H2 be the values of 47rinll and 47ri' - 'Z/ l for the 2 2 same induction B, it can be shown that the true magnetizing force is H = H l - (H 2 - H 1).

The equation F = B 2 /87r is often said to express " Maxwell's law of magnetic traction " (Maxwell, Electricity and Magnetism,, §§ 642-646).

The values assigned to H were calculated from H= 2ni/r, and ranged from 3.9 to 585, but inasmuch as no account was taken of any 2 Since in most practicable experiments H 2 is negligible in comparison with B 2, the force may be taken as B 2 /87r without sensible= error.

He showed that there was, on the whole, a fair agreement between the values determined ballistic ally and those given by the formula B = 871-F.

If the pull is measured in pounds and the area in square inches, the formula may be written B =1317 X iI P/S +H.

The test-piece A, surrounded by a magnetizing coil, is clamped between two soft-iron blocks B, B'.

The yoke has two projecting pieces C, C' at unequal distances from the knife-edges, and separated from the blocks B, B' by narrow air-gaps.

The weight W is moved along the scale until the yoke just tilts over upon the stop S; the distance of W from its zero position is then, as can easily be shown, proportional to F, and therefore to B 2, and approximately to I 2.

Let m be the number of turns in use, and H 1 and H2 the magnetizing forces which produce the same induction B in the test and the standard rods respectively; then H1=H2Xm/Ioo.

The value of B which corresponds to Hem/100 can be found from the FIG.

Thus any desired number of corresponding values of H and B can be easily and quickly found.

Values of I are derived from (B -H)/477and of from B/H.

There appears to be no definite limit to the value to which the induction B may be raised, but the magnetization I attains a true saturation value under magnetizing forces which are in most cases comparatively moderate.

When the saturation value of I has been reached, the relation of magnetic induction to magnetic force may be expressed by B = H +constant.

It is shown in the paper that the greatest possible force which the isthmus method can apply at a point in the axis of the bobbin is F = 11, 137 I, log i n b/a, I, being the saturation value of the magnet pores, a the radius of the neck on which the cones converge, and b the radius of the bases of the cones.

If a long magnetized rod is divided transversely and the cut ends placed nearly in contact, the magnetic force inside the narrow air gap will be B = H +47rI.

The force acting on the magnetism of one of the faces, and urging this face towards the other, will be less than B by 27r1, the part of the total force due to the first face itself; hence the force per unit of area with which the faces would press against each other if in contact is P = (B-27rI)I =27rT 2 +HI = (B 2 -H 2) =/81r.

The whole of the rod will therefore be subject to a compressive longitudinal stress P, the associated contraction R, expressed as a fraction of the original length, being R = P/M = (B 2 -H2)/87-M, where M is Young's modulus.

Those who support this view generally speak of the stress as " Maxwell's stress," and assume its value to be B 2 /87r.

Further, Maxwell's stress is a tension along the lines of force, and is equal to B 2 /87r only when B = H, and there is no magnetization.

Let each of the effects A, B, C, D and E be called positive when it is such as is exhibited by moderately magnetized iron, and negative when its sense is opposite.

B r (S) = strongly magnetized.

The range of + B within which Steinmetz's formula is applicable becomes notably increased at low temperature.

B =9000, at a frequency of loo per second, was only 0.254 watt per pound, as compared with 0.382 for the Swedish iron.

Its hysteresis-loss for B =9000 was o 236 per pound.

In one case the hysteresis loss per cubic centimetre per cycle was 16,100 ergs for B =1 5,900, and only 1200 ergs for B = 20,200, the highest induction obtained in the experiment; possibly it would have vanished before B had reached 21,000.2 These experiments prove that actual friction must be almost entirely absent, and, as Baily remarks, the agreement of the results with the previously suggested deduction affords a strong verification of Ewing's form of the molecular theory.

To obtain the support of the capitalists, Gaius Gracchus conceived the plan of creating friction between them and the senate, which he carried out by handing over to them the control (a) of the jury-courts, and (b) of the revenues of Asia.

B, Diagram showing the nature of this infolding.

B, Rhabdom of the same, consisting of five confluent rhabdomeres.

The figure B also shows the peculiar neural investiture formed by the cerebral arteries in Limulus and the derivation from this of the arteries to the limbs, III, IV, VI, whereas in Scorpio the latter have a separate origin from the anterior aorta.

Whilst the chances are in favour of some one homoplastic coincidence or structural agreement occurring between some member or other of a large group a and some member or other of a large group b, the matter is very different 2 a o FIG.

A, Youngest stage with no mesosomatic somites; B and C, stages with two mesosomatic somites between the prosomatic and telsonic carapaces; D, adult condition, still with only two free mesosomatic somites.

In front of it the narrow waist is formed by the soft sternal area of the praegenital somite; 2, the sternite of the 2 second opisthosomatic somite covering the posterior pair of lung-sacs; and 4, the spinning appendages (limbs) of the opisthosoma; a, inner, b, outer ramus of the appendage; I I, sternite of the eleventh --

Opisthosoma consisting of only ten somites, which have no tergal and sternal elements, the prae-genital somite contracted to form a " waist," as in the Pedipalpi; the last three narrowed to form a A B prae-1 2345 6789 io I I111I IV V VI gen Opisttaosoma Prosoma FIG.

A, Ventral view of prosoma and of anterior region of opistho soma with the appendages cut off near the base; a and b, B, Dorsal view.

The segmentation of the prosoma and the form of the appendages bear a homoplastic similarity to the head, pro-, meso-, and meta-thorax of a Hexapod with mandibles, maxillary palps and three pairs of walking legs; while the opistho io i e d c b o a S' S" 2 I VT V S IV III II I Opisthosoma Prosoma FIG.

Tofieldia, an arctic and alpine genus of small herbs with a slender scape springing from a tuft of narrow ensiform leaves and bearing a raceme of small green flowers; Narthecium (bog-asphodel), herbs with a habit similar to Tofieldia, but with larger golden-yellow flowers; and Colchicum, a genus with about 30 species including b the meadow saffron or autumn crocus (C. autumnale).

He not only freed it from all trammels of geometrical construction, but by the introduction of the symbol b gave it the efficacy of a new calculus.

To group (a) belong territorial differences in regard to ownership of land and rights of fishing at sea; to group (b) belong pecuniary claims in respect of acts wrongfully done to one or more subjects of one state by, or with the authority of, another state.

Of a more distinctly national tendency are the lyrics of John Kriza b and John Erdelyi, but the reputation of the latter was more especially due to his collections of folk-lore made on behalf of the Kisfaludy society.

To say, for instance, that X is equal to A -B, is the same thing as to say that X is a quantity such that X and B, when added, make up A; and the above five statements of necessary connexion between two statements of equality are in fact nothing more than definitions of the symbols -, m of, =,, and loga.

If, for instance, we state that A=X - B, this is really a statement that X=Ad-B.

When we are familiar with the treatment of quantities by equations, we may ignore the units and deal solely with numbers; and (ii.) (a) and (ii.) (b) may then, by the commutative law for multiplication, be regarded as identical.

In the case of addition, for instance, suppose that we are satisfied that in a+b+c+d+e we may take any two, as b and c, together (association) and interchange them (commutation).

Let X and Y be the related quantities, their expressions in terms of selected units A and B being x and y, so that X=x.A, Y = y.

An equation of the form ax=b, where a and b do not contain x, is the standard form of simple equation.

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

If a of the stock are of one type, b of another, c of another,.

Suppose that there are a number of arrangements of r terms or elements, the first of which a is always either A or not-A, the second b is B or not-B, the third c is C or not-C, and so on.

If, in this latter case, the proportion of cases in which b is B to cases in which b is not-B is the same for the group of pN individuals in which a is A as for the group of (I-p)N in which a is not-A, then the frequencies of A and of B are said to be independent; if this is not the case they are said to be correlated.

If the frequency of each A is independent of the frequency of each B, then the attributes a and b are independent; otherwise they are correlated.

The values of the first ten of Bernoulli's numbers are B1= t, B2= 1, B3 =412, B4 =30, B5 =6 = 6 9 1 B7 = l, B =3 =4 4 fl, IV.

The results in (b) apply also if n is a negative integer.

We want to express (23/13) 3 in the form a l b, where b is nearly equal to i.

It could not escape notice that one and the same symbol, such as -1 (a - b), or even (a - b), sometimes did and sometimes did not admit of arithmetical interpretation, according to the values attributed to the letters involved.

They are (a+b)-?-c=a+(b+c) (A) (aXb)Xc=aX(bXc) (A') a+b=b+a (c) aXb=bXa (c') a(b c) =ab-Fac (D) (a - b)+b=a (I) (a=b)Xb=a (I') These formulae express the associative and commutative laws of the operations + and X, the distributive law of X, and the definitions of the inverse symbols - and =, which are assumed to be unambiguous.

Let i denote a definite region of space; and let a, b, &c., stand for definite parts of i.

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 can be proved by geometry that (aA-H3B) +yC = aA+(aB+- y C) = (a + 1 3+ 7) P, where P is in fact the centroid of masses a, 13, y placed at A, B, C respectively.

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

From A Merely Formal Point Of View, We Have In The Barycentric Calculus A Set Of " Special Symbols Of Quantity " Or " Extraordinaries " A, B, C, &C., Which Combine With Each Other By Means Of Operations And Which Obey The Ordinary Rules, And With Ordinary Algebraic Quantities By Operations X And =, Also According To The Ordinary Rules, Except That Division By An Extraordinary Is Not Used.

If A 1 = X a i e i, B i = /if i e i, the distributive law of multiplication is preserved by assuming A1B1=E(a0 i 3)eiej; it follows that A 1 B 1 = - B 1 A 1, and that A l 2 = o.

The symbol AB, from this point of view, is called the inner product of A and B, as distinguished from the outer product IAB.

The outer and inner products of two extensive quantities A, B, are in many ways analogous to the quaternion symbols Vab and Sab respectively.

These may be compared and contrasted with such quaternion formulae as S(VabVcd) =SadSbc-SacSbd dSabc = aSbcd - bScda+cSadb where a, b, c, d denote arbitrary vectors.

One of the most important questions in investigating a linear algebra is to decide the necessary relations between a and b in order that this product may be zero.

Even in ordinary algebra the notation for powers and roots disturbs the symmetry of the rational theory; and when a schoolboy illegitimately extends the distributive law by writing -V (a+b)a+J b, he is unconsciously emphasizing this want of complete harmony.

Study), and B I c (G.

In this they were completely successful, for they obtained general solutions for the equations ax by = c, xy = ax+by+c (since rediscovered by Leonhard Euler) and cy 2 = ax e + b.

The Inter-Allied Commission entrusted with the details was ordered to divide the disputed area into Zone A, mainly south of the river Drava (Drau) and Zone B,.

Zone B, with Klagenfurt, now automatically passed to Austria.

Against Bulgaria the Yugoslav delegation claimed considerable frontier rectifications - (a) the Strumnica salient, which threatened the Vardar railway from the east, (b) the district of Kochana (Tocana) and the Bregalnitsa (Bregalnica), (c) a strip of territory running parallel with the old Serbo-Bulgarian frontier the whole way from Zajecar to Kyustendil, and (d) the town of Vidin on the Danube and the salient between it and the Timok.

Pre-Darwinian zoologists had been aware of the class of facts thus interpreted by Fritz Muller, but the authoritative view on the subject had been that there is a parallelism between (a) the series of forms which occur in individual development, (b) the series of existing forms from lower to higher, and (c) the series of forms which succeed 'one another in the strata of the earth's crust, whilst an explanation of this parallelism was either not attempted, or was illusively offered in the shape of a doctrine of harmony of plan in creation.

The most important work in this direction has been done by Fritz Muller (Filr Darwin), by Herman Muller (Fertilization of Plants by Insects), Grade b.

We will now apply the integrals (2) to the case of a rectangular aperture of width a parallel to x and of width b parallel to y.

The form of (3) shows immediately that, if a and b be altered, the co-ordinates of any characteristic point in the pattern vary as a-'- and b-1.

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

Thus if A be selfluminous, the illumination is a maximum at B, where all the secondary waves agree in phase.

When the interval is very small the discrepancy, though mathematically existent, produces no practical effect, and the illumination at B due to P is as important as that due to A, the intensities of the two luminous sources being supposed equal.

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

The illumination at B due to P then becomes comparatively small, indeed for some forms of aperture evanescent.

The extreme discrepancy is that between the waves which travel through the outermost parts of the object-glass at L and L'; so that if we adopt the above standard of resolution, the question is where must P be situated in order that the relative retardation of the rays PL and PL' may on their arrival at B amount to a wave-length (X).

The 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.

Thus f t2 ds (for A) = f A ds (for B)..

The path of a ray from the wave-surface A 0 B 0 to the point A is changed; but in virtue of the minimum property the change may be neglected in calculating the optical distance,as it influences the result by quantities of the second order only in the changes of refrangibility.

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.

These directions, in which the retardation between A and B is exactly mnX, may be called the principal directions.

Then the relative retardation of the extreme rays (corresponding to the edges A, B of the grating) is mnX.

B is the point at which the effect is required, distant a+b from 0, so that AB= b, AP=s, PQ ds.

When a, b, X are regarded as constant, the first factor may be omitted, - as indeed should be done for consistency's sake, inasmuch as other factors of the same nature have been omitted already.

The 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?

By means of this expression we may trace the locus of a band of given order as b varies.

The occurrence of sin 4 as a factor in (6) shows that the relative intensities of the primary light and of that diffracted in the direction B depend upon the condition of the former as regards polarization.

The force operative upon the positive half is parallel to OZ, and of amount per unit of area equal to - b 2 D = b 2 kD cos nt; and to this force acting over the whole of the plane the actual motion on the positive side may be conceived to be due.

At B, where the ray leaves the drop, the deviation is the same as at A, viz.

There it was that the community formed by Yahya b.

The effect of overwork upon an organ or tissue varies in accordance with (a) the particular organ or tissue concerned, (b) the amount of nourishment conveyed to it, and (c) the power of assimilation possessed by its cells.

Its scope may be briefly indicated as including (a) duties exercised elsewhere by the Borough Councils, and by the London County Council (although that body is by no means powerless within the City boundaries); and (b) peculiar duties such as control of markets and police.

With the exception of the townships and a district of Emtonjaneni magistracy known as " Proviso B," 1 mainly occupied by Boer farmers, all the land was vested in the crown and very little has been parted with to Europeans.

Two of these (UBI, L B 15) were attached to the Austrian submarine force.

A and B are pear-shaped glass vessels connected by a long narrow india-rubber tube, which must be sufficiently strong in the body (or strengthened by a linen coating) to stand an outward pressure of 1 to 2 atmospheres.

The apparatus, after having been carefully cleaned and dried, is charged with pure and dry mercury which must next be worked backwards and forwards between A and B to remove all the air-bells.

It can be considerably shortened, the two vessels A and B brought more closely together, and the somewhat objectionable india-rubber tube be dispensed with, if we connect the air-space in B with an ordinary air pump, and by means of it do the greater part of the sucking and the whole of the lifting work.

It is expressly directed by the act of 1898 above referred to, that in regard to succession, inheritance, marriage, caste or any religious usage or institution, the law to be administered in Burma is (a) the Buddhist law in cases where the parties are Buddhists, (b) the Mahommedan law in cases where the parties are Mahommedans, (c) the Hindu law in cases where the parties are Hindus, except so far as the same may have been modified by the legislature.

His elder brother Ram Rai was passed over p was put to death for refusal to embrace Islam b.

A group (B), comprising copper, is, substantially, attacked only in the presence of oxygen or air.

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.

This is proved by taking the two points A and B in the same vertical line, and considering the equilibrium of the prism by resolving vertically.

In this case the thrust at the lower end B must exceed the thrust at A, the upper end, by the weight of the prism of liquid; so that, denoting the cross section of the prism by a ft.

Afterwards, when the metal has risen above B, to the level KK', the additional thrust is the weight of the cylinder of diameter KK' and height BH.

For if the ship turns through a small angle 0 about the line FF', then b1, b 2, the C.G.

Ballasted with G at B, the righting couple when the ship is heeled through 0 is given by W.BM.

The time rate of increase of momentum of the fluid inside S is )dxdydz; (5) and (5) is the sum of (I), (2), (3), (4), so that /if (dpu+dpu2+dpuv +dpuw_ +d p j d xdyd z = o, (b)` dt dx dy dz dx / leading to the differential equation of motion dpu dpu 2 dpuv dpuv _ X_ (7) dt + dx + dy + dz with two similar equations.

Taking the axis of x for an instant in the normal through a point on the surface H = constant, this makes u = o, = o; and in steady motion the equations reduce to dH/dv=2q-2wn = 2gco sin e, (4) where B is the angle between the stream line and vortex line; and this holds for their projection on any plane to which dv is drawn perpendicular.

If the direction of motion makes an angle 0' with Ox, tan B' = d0 !dam _ ?xy 2 = tan 20, 0 =-10', (9) dy/ y and the velocity is Ua2/r2.

When the cylinder r =a is moved with velocity U and r =b with velocity U 1 along Ox, = U b e - a,1 r +0 cos 0 - U ib2 - 2 a, (r +Q 2 ') cos 0, = - U be a2 a2 (b 2 - r) sin 0 - Uib2 b1)a, (r - ¢2 sin 0; b and similarly, with velocity components V and V 1 along Oy a 2 b2 ?= Vb,_a,(r+r) sin g -Vi b, b2 a, (r+ 2) sin 0, (17) = V b, a2 a, (b2 r) cos 0+Vi b, b, a, (r- ¢ 2) cos h; (18) and then for the resultant motion z 2zz w= (U 2 + V2)b2a a2U+Vi +b a b a2 U z Vi -(U12+V12) b2 z a2b2 Ui +VIi b 2 - a 2 U1 +Vii b 2 - a 2 z The resultant impulse of the liquid on the cylinder is given by the component, over r=a (§ 36), X =f p4 cos 0.ad0 =7rpa 2 (U b z 2 + a 2 Uib.2bz a2); (20) and over r =b Xi= fp?

Taking two planes x = =b, and considering the increase of momentum in the liquid between them, due to the entry and exit of liquid momentum, the increase across dy in the direction Oy, due to elements at P and P' at opposite ends of the diameter PP', is pdy (U - Ua 2 r2 cos 20 +mr i sin 0) (Ua 2 r 2 sin 2 0+mr 1 cos 0) + pdy (- U+Ua 2 r 2 cos 2 0 +mr1 sin 0) (Ua 2 r 2 sin 2 0 -mr 1 cos 0) =2pdymUr '(cos 0 -a 2 r 2 cos 30), (8) and with b tan r =b sec this is 2pmUdo(i -a 2 b2 cos 30 cos 0), (9) and integrating between the limits 0 = 27r, the resultant, as before, is 27rpmU.

The velocity of a liquid particle is thus (a 2 - b 2)/(a 2 +b 2) of what it would be if the liquid was frozen and rotating bodily with the ellipse; and so the effective angular inertia of the liquid is (a 2 -b 2) 2 /(a 2 +b 2) 2 of the solid; and the effective radius of gyration, solid and liquid, is given by k 2 = 4 (a 2 2), and 4 (a 2 For the liquid in the interspace between a and n, m ch 2(0-a) sin 2E 4) 1 4Rc 2 sh 2n sin 2E (a2_ b2)I(a2+ b2) = I/th 2 (na)th 2n; (8) and the effective k 2 of the liquid is reduced to 4c 2 /th 2 (n-a)sh 2n, (9) which becomes 4c 2 /sh 2n = s (a 2 - b 2)/ab, when a =00, and the liquid surrounds the ellipse n to infinity.

Example 3.-Analysing in this way the rotation of a rectangle filled with liquid into the two components of shear, the stream function 1//1 is to be made to satisfy the conditions v 2 /1 =0, 111+IRx 2 = IRa 2, or /11 =o when x= = a, +b1+IRx 2 = I Ra2, y ' 1 = IR(a 2 -x 2), when y = b Expanded in a Fourier series, 2 232 2 cos(2n+ I)Z?rx/a a -x 7r3 a Lim (2n+I) 3 ' (1) so that '?"

In the absolute path in space cos Ili = (2 - 3 sin 2 6)/1/ (4-sin 2 6), and sin 3 B = (y 3 -c 2 y)/a 3, (19) which leads to no simple relation.

As an application of moving axes, consider the motion of liquid filling the ellipsoidal case 2 y 2 z2 Ti + b1 +- 2 = I; (1) and first suppose the liquid be frozen, and the ellipsoid l3 (4) (I) (6) (9) (I o) (II) (12) (14) = 2 U ¢ 2, (15) rotating about the centre with components of angular velocity, 7 7, f'; then u= - y i +z'i, v = w = -x7 7 +y (2) Now suppose the liquid to be melted, and additional components of angular velocity S21, 522, S23 communicated to the ellipsoidal case; the additional velocity communicated to the liquid will be due to a velocity-function 2224_ - S2 b c 6 a 5 x b2xy, as may be verified by considering one term at a time.

If there are more B corners than one, either on xA or x'A', the expression for i is the product of corresponding factors, such as in (5) Restricting the attention to a single corner B, a = n(cos no +i sin 110) _ (b-a'.0-a) +1!

Ja - u  ?I a -a b -u' sh nS2=sh log (Q)=?a - b a - a' b - u' At x where = co, u = o, and q= go, (O n b - a ' a + a -b a' cio) - ?a-a'?b a-a' q In crossing to the line of flow x'A'P'J', b changes from o to m, so that with q = Q across JJ', while across xx the velocity is qo, so that i n = go.

Helmholtz, of the efflux of a jet between two edges A and A 1 in an infinite wall, is obtained by the symmetrical duplication of the above, with n = 1, b = o, a' = - oo, as in fig.

Generally, by making a' = -oo, the line x'A' may be taken as a straight stream line of infinite length, forming an axis of symmetry; and then by duplica tion the result can be ob A tained, with assigned n, a, and b, of the efflux from a symmetrical converging FIG.

Thus with a' =o, a stream is split symmetrically by a wedge of angle ' zr/n as in Bobyleff's problem; and, by making a = oo, the wedge extends to infinity; then chnS2= u, sh nS2= b n u.

Over the ellipsoid, p denoting the length of the perpendicular from the centre on a tangent plane, px _ pv _ _ pz 1= a2+X' b +A' n c2+A p2x2 + p2y2 p2z2 I (a2 - + X)2 (b 2 +x)2 + (0+X)2, p 2 = (a2+A)12+(b2+X)m2+(c2+X)n2, = a 2 1 2 +b 2 m 2 +c 2 n 2 +X, 2p d = ds; (8) Thence d?

When the liquid is bounded externally by the fixed ellipsoid A = A I, a slight extension will give the velocity function 4 of the liquid in the interspace as the ellipsoid A=o is passing with velocity U through the confocal position; 4 must now take the formx(1'+N), and will satisfy the conditions in the shape CM abcdX ¢ = Ux - Ux a b x 2+X)P Bo+CoB I - C 1 (A 1 abcdX, I a1b1cl - J o (a2+ A)P and any'confocal ellipsoid defined by A, internal or external to A=A 1, may be supposed to swim with the liquid for an instant, without distortion or rotation, with velocity along Ox BA+CA-B 1 -C1 W'.

The extension to the case where the liquid is bounded externally by a fixed ellipsoid X= X is made in a similar manner, by putting 4 = x y (x+ 11), (io) and the ratio of the effective angular inertia in (9) is changed to 2 (B0-A0) (B 1A1) +.a12 - a 2 +b 2 a b1c1 a -b -b12 abc (Bo-Ao)+(B1-A1) a 2 + b 2 a1 2 + b1 2 alblcl Make c= CO for confocal elliptic cylinders; and then _, 2 A? ?

Sources of Apocalyptic. - The origin of Apocalyptic is to be sought in (a) unfulfilled prophecy and in (b) traditional elements drawn from various sources.

An /,, B other method is to boil one FIG.

A and B, mycelium (m), the mix t ure over the affe c ted with haustoria (h).

Extraction of cane juice by diffusion (a process more fully described under the head of beetroot sugar manufacture) is adopted in a few plantations in Java and Cuba, in Louisiana Etr cti o n and the Hawaiian Islands, and in one or two factories y f i in Egypt; b u t hitherto, except under exceptional conditions (as at Aska, in the Madras Presidency, where the local price for sugar is three or four times the London price), it would not seem to offer any substantial advantage over double or triple crushing.

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

B, Archigetes sieboldi; X 60 (from Leuckart).