T31  = (Ql — 0.2+0'2 — 2'3) 2To =T32+T23+2(Ql—Q2) (0-2— a3) To ; so that T31 necessarily exceeds the sum of the other two inter-facial tensions . We are thus led to the important conclusion that according to this See also:

• HYPOTHESIS (from Gr. inrorcOivat, to put under; cf. Lat. suppositio, from sub-ponere)

hypothesis See also:

• NEUMANN, FRANZ ERNST (1798-1895)
• NEUMANN, KARL FRIEDRICH (1793-1870)

Neumann's triangle is necessarily imaginary, that one of three fluids will always spread upon the interface of the other two . Another point of importance may be easily illustrated by this theory, viz. the dependency of capillarity upon abruptness of transition . " The See also:

• REASON (Lat. ratio, through French raison)

reason why the capillary force should disappear when the transition between two liquids is sufficiently See also:

• GRADUAL (Med. Lat. gradualis, of or belonging to steps or degrees; gradus, step)

gradual will now be evident . Suppose that the transition from o to o is made in two equal steps, the thickness of the inter-mediate layer of See also:

• DENSITY (Lat. densus, thick)

density Zo- being large compared to the range of the molecular forces, but small in comparison with the See also:

• RADIUS

radius of curvature . At each step the difference of capillary pressure is only one-See also:

• QUARTER (through Fr. from Lat. quartarius, fourth part)

quarter of that due to the sudden transition from o to a, and thus altogether See also:

• HALF

half the effect is lost by the inter-position of the layer . If there were three equal steps, the effect would be reduced to one-third, and so on . When the number of steps is See also:

• INFINITE (from Lat. in, not, finis, end or limit; cf. findere, to deave)

infinite, the capillary pressure disappears altogether." (" See also:

• LAPLACE, PIERRE SIMON, MARQUIS DE (1749—1827)

Laplace's Theory of Capillarity," See also:

• RAYLEIGH, JOHN WILLIAM STRUTT

Rayleigh, Phil . Mag., 1883, P . 315.) According to Laplace's hypothesis the whole See also:

• ENERGY (from the Gr. ivipyela; iv, in, ipyov, work)

energy of any number of contiguous strata of liquids is least when they are arranged in See also:

• ORDER
• ORDER (through Fr. ordre, for earlier ordene, from Lat. ordo, ordinis, rank, service, arrangement; the ultimate source is generally taken to be the root seen in Lat. oriri, rise, arise, begin; cf. " origin ")
• ORDER, HOLY

order of density, so that this is the disposition favoured by the attractive forces . The problem is to make the sum of the interfacial tensions a minimum, each tension being proportional to the square of the difference of densities of the two contiguous liquids in question . If the order of stratification differ from that of densities, we can show that each step of approximation to this order lowers the sum of tensions .

To this end consider the effect of the abolition of a stratum o n+l, contiguous to o-n and 0-n+2 . Before the See also:

• CHANGE (derived through the Fr. from the Late Lat. cambium, cambiare, to barter; the ultimate derivation is probably from the root which appears in the Gr. «a urmu', to bend)

change we have (a.—o-n+1)2+(fin+1—o•n+2)2, and afterwards (Un The second minus the first, or the increase in the sum of tensions, is thus 2 (Qn — Qn+l) (Qn+l — Qn+2) Hence, if o-n+1 be intermediate in magnitude between an and an+2, the sum of tensions is increased by the abolition of the stratum; but, if trn+1 be not intermediate, the sum is decreased . We see, then, that the removal of a stratum from between neighbours where it is out of order and its introduction between neighbours where it will be in order is doubly favourable to the reduction of the sum of tensions; and since by a See also:

• SUCCESSION (Lat. successio, from succedere, to follow after)

succession of such steps we may arrive at the order of magnitude through-out, we conclude that this is the disposition of minimum tensions and energy . So far the results of Laplace's hypothesis are in marked accordance with experiment; but if we follow it out further, discordances begin to See also:

• MANIFEST (Lat. manifestus, clear, open to view)

manifest themselves . According to (52) l T31 = See also:

• T12

T12+ V T23, (53) a relation not verified by experiment . What is more, (52) shows that according to the hypothesis T12 is necessarily See also:

• POSITIVE (or PORTABLE) ORGAN

positive; • • • (47) 2 which See also:

• YOUNG
• YOUNG, A
• YOUNG, ARTHUR (1741-1820)
• YOUNG, BRIGHAM (1801-1877)
• YOUNG, CHARLES MAYNE (1777–1856)
• YOUNG, EDWARD (1683–1765)
• YOUNG, JAMES (1811-1883)
• YOUNG, THOMAS (1773-1829)

Young probably had in view, namely that the force in each See also:

• CASE
• CASE, JOHN (d. 1600)

case was See also:

• CONSTANT, BENJAMIN (1845-1902)

constant within a limited range, the same in all cases, and vanished outside that range . As an immediate consequence of this hypothesis we have from (28) so that, if the preceding See also:

• ARGUMENT

argument be correct, no such thing as mixture of two liquids could ever take See also:

• PLACE (through Fr. from Lat. platea, street; Gr. IrAar6s, wide)

place . There are two apparent exceptions to Marangoni's See also:

• RULE

rule which See also:

• CALL (from Anglo-Saxon ceallian, a common Teutonic word, cf. Dutch kallen, to talk or chatter)

call for a word of explanation . According to the rule, See also:

• WATER

water, which has the See also:

• LOWER

lower See also:

• SURFACE

surface-tension, should spread upon the surface of See also:

• MERCURY
• MERCURY (MERCU1uus)
• MERCURY (symbol Hg, atomic weight = 2oo)

mercury; whereas the universal experience of the laboratory is that drops of water See also:

• STANDING

standing upon mercury retain their compact See also:

• FORM (Lat. forma)

form without the least tendency to spread . To Quincke belongs the See also:

• CREDIT (Lat. credere, to believe)

credit of dissipating the apparent exception . He found that mercury specially prepared behaves quite differently from See also:

• ORDINARY (med. Lat. ordinarius, Fr. ordinaire)

ordinary mercury, and that a drop of water deposited thereon spreads over the entire surface . The ordinary behaviour is evidently the result of a film of grease, which adheres with See also:

• GREAT

great obstinacy .

The See also:

• PROCESS

process described by Quincke is somewhat elaborate; but there is little difficulty in repeating the experiment if the See also:

• MISTAKE (i.e. take amiss)

mistake be avoided of using a See also:

• FREE

free surface already contaminated, as almost inevitably happens when the mercury is poured from an ordinary See also:

• BOTTLE (Fr. bouteille, from a diminutive of the Lat. butta, a flask; cf. Eng. " butt ")

bottle . The mercury should be See also:

• DRAWN

drawn from underneath, for which purpose an arrangement similar to a chemical See also:

• WASH, THE

wash bottle is suitable, and it may be poured into See also:

• WATCH (in O. Eng. wcecce, a keeping guard or watching, from wacian, to guard, watch, wacan, to wake)

watch-glasses, previously dipped into strong sulphuric See also:

• ACID (from the Lat. root ac-, sharp; acere, to be sour)

acid, rinsed in distilled water, and dried over a See also:

• BUNSEN, CHRISTIAN CHARLES JOSIAS, BARON VON (1791-186o,)
• BUNSEN, ROBERT WILHELM VON (1811-1899)

Bunsen See also:

• FLAME (Lat. flamma; the root flag- appears in flagrare, to burn, blaze, and Gr. d, XEryew)

flame . When the glasses are cool, they may be charged with mercury, of which the first See also:

• PART

part is rejected . Operating in this way there is no difficulty in obtaining surfaces upon which a drop of water spreads, although from causes that cannot always be traced, a certain proportion of failures is met with . As might be expected, the grease which produces these effects is largely volatile . In many cases a very moderate preliminary warming of the watch-glasses makes all the difference in the behaviour of the drop . The behaviour of a drop of See also:

• CARBON (symbol C, atomic weight 12)

carbon bisulphide placed upon clean% water is also, at first sight, an exception to Marangoni's rule . So far from spreading over the surface, as according to its lower surface-tension it ought to do, it remains suspended in the form of a See also:

• LENS
• LENS (from Lat. lens, lentil, on account of the similarity of the form of a lens to that of a lentil seed)

lens . Any dust that may be lying upon the surface is not driven away to the edge of the drop, as would happen in the case of oil . A See also:

• SIMPLE

simple modification of the experiment suffices, however, to clear up the difficulty . If after the deposition of the drop, a little See also:

• LYCOPODIUM

lycopodium be scattered over the surface, it is seen that a circular space surrounding the drop, of about the See also:

• SIZE

size of a See also:

• SHILLING

shilling, remains See also:

• BARE

bare, and this, however often the dusting be repeated, so See also:

• LONG, GEORGE (1800-1879)
• LONG, JOHN DAVIS (1838– )

long as any of the carbon bisulphide remains . The See also:

• INTERPRETATION (from Lat. interpretari, to expound, explain, inter pres, an agent, go-between, interpreter; inter, between, and the root pret-, possibly connected with that seen either in Greek 4 p4'ew, to speak, or irpa-rrecv, to do)

interpretation can hardly be doubtful .

The carbon bisulphide is really spreading all the while, but on See also:

• ACCOUNT
• ACCOUNT (through O. Fr. acont, Late Lat. comptum, cornputare, to calculate)

account of its volatility is unable to reach any considerable distance . Immediately surrounding the drop there is a film moving outwards at a high See also:

• SPEED, JOHN (1552–1629)

speed, and this carries away almost instantaneously any dust that may fall upon it . The phenomenon above described requires that the water-surface be clean . If a very little grease be See also:

• PRESENT

present, there is no outward flow and dust remains undisturbed in the immediate neighbourhood of the drop.] On the Rise of a Liquid in a See also:

• TUBE (Lat. tuba)

Tube.—Let a tube (fig . 6) whose See also:

• INTERNAL

internal radius is r, made of a solid substance c, be dipped into T T a liquid a . Let us suppose b c that the See also:

• ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

angle of contact for this liquid with the solid c is an acute angle . This implies that the ten- See also:

• SION [Ger. Sitten]

sion of the free surface of the solid c is greater than that of the surface of contact of the solid with the liquid a . Now See also:

• CON

con- sider the tension of the free surface of the liquid a . All See also:

• ROUND (O. Fr. rond, Lat. rotundus, the Fr. is the source also of Du. rond; Ger., Swed., Dan. and Nor. rond)

round its edge there is a tension T acting at an angle a with the See also:

• VERTICAL (from Lat. vertex, highest point)

vertical . The circumference of the edge is 27rr, so that the resultant of this tension is a force 27rrT See also:

• COS
• COS, or STANKO (Ital. Stanchio, Turk. Islan-keui, by corruption from Etc rav KW)

cos a acting vertically upwards on the liquid . Hence the liquid will rise in the tube till the See also:

• WEIGHT

weight of the vertical See also:

• COLUMN (Lat. columna)

column between the free surface and the level of the liquid in the See also:

• VESSEL (O. Fr. vaissel, from a rare Lat. vascellum, dim. of vas, vase, urn)

vessel balances the resultant of the surface-tension . The upper surface of this column is not level, so that the height of the column cannot be directly measured, but let us assume that h is the mean height of the column, that is to say, the height of a column of equal weight, but with a See also:

• FLAT (a modification of O. Eng. flet, an obsolete word of Teutonic origin, meaning the ground beneath the feet)

flat See also:

• TOP (cf. Dan. top, Ger. Topf, also meaning pot)

top .

Then if r is the radius of the tube at the top of the column, the See also:

• VOLUME

volume of the suspended column is 7rr2h, and its weight is 7rpgr2h, when p is its density and g the intensity of gravity . Equating this force with the resultant of the tension ,rpgr2h = 2,rrT cos a, or h= 2T cos a/pgr . Hence the mean height to which the fluid rises is inversely as the radius of the tube . For water in a clean See also:

• GLASS
• GLASS (O.E. glees, cf. Ger. Glas, perhaps derived from an old Teutonic root gla-, a variant of glo-, having the general sense of shining, cf. " glare," " glow ")
• GLASS, STAINED

glass tube the angle of contact is zero, and h=2T/pgr . For mercury in a glass tube the angle of contact is 128° 52', the cosine of which is negative . Hence when a glass tube is dipped into a vessel of mercury, the mercury within the tube stands at a lower level than outside it . Rise of a Liquid between Two Plates.—When two parallel plates are placed vertically in a liquid the liquid rises between them . If we now suppose fig . 6 to represent a vertical See also:

• SECTION
• SECTION (Lat. sectio, cutting, secare, to cut)

section perpendicular to the plates, we may calculate the rise of the liquid . Let l be the breadth of the plates measured perpendicularly to the See also:

• PLANE

plane of the See also:

• PAPER
• PAPER (Fr. papier, from Lat. papyrus)

paper, then the length of the See also:

• LINE

line which See also:

• BOUNDS, BEATING THE

bounds the wet and the dry parts of the plates inside is 1 for each surface, and on this the tension T acts at an angle a to the vertical . Hence the resultant of the surface-tension is 21 T cos a . If the distance between the inner surfaces of the plates is a, and if the mean height of the film of fluid which rises between them is h, the weight of fluid raised is pghla .

Equating the forces pghla = 21T cos a, whence h =2T cos a/pga . This expression is the same as that for the rise of a liquid in a tube, except that instead of r, the radius of the tube, we have a the distance of the plates . Form of the Capillary Surf See also:

• ACE (derived through the Lat. as, from the Tarentine form of the Gr. etc)

ace.—The form of the surface of a liquid acted on by gravity is easily determined if we assume that near the part considered the line of contact of the surface of the liquid with that of the solid bounding it is straight and See also:

• HORIZONTAL

horizontal, as it is when the solids which constrain the liquid are bounded by surfaces formed by horizontal and parallel generating lines . This will be the case, for instance, near a flat See also:

• PLATE

plate dipped into the liquid . If we suppose these generating lines to be normal to the plane of the paper, then all sections of the solids parallel to this plane will be equal and similar to each other, and the section of the surface of the liquid will be of the same form for all such sections . Let us consider the portion of the liquid between two parallel sections distant one unit of length . Let PI, P2 (fig . 7) be two points of the surface; B,, 02 the inclination of the surface 2 to the See also:

• HORIZON (Gr. 6pfTwv, dividing)

horizon at P, and P2; Y1, y2 the heights of PI and P2 above the level of the Iiquid at a distance from all solid bodies . The pressure at any point of the liquid which is above this level is negative Ti unless another fluid as, for in-stance, the See also:

• AIR (from an Indo-European root meaning " breathe," " blow ")
• AIR, or ASBEN

air, presses on the upper surface, but it is only the difference of pressures with which we have to do, because two equal pressures on opposite sides of the surface produce no effect . We may, therefore, write for the pressure at a height y 11= — pgY A where p is the density of the liquid, or if there are two fluids the excess of the density of the lower fluid over that of the upper one . The forces acting on the portion of liquid P1P2A2A1 are—first, the horizontal pressures, -2pgyi and 1pgy2; second, the surface-tension T acting at P1 and P2 in directions inclined Ui and 02 to the horizon . Resolving horizontally we find T(cos 02— cos 01)+1gp(y22—y12) =o, whence cos 02 = cos o, +gpy12 — gpy22 2T 2T or if we suppose P1 fixed and P2 variable, we may write cos O = constant — zgpy2/T .

This See also:

• EQUATION (from Lat. aequatio, aequare, to equalize)

equation gives a relation between the inclination of the See also:

• CURVE (Lat. curvus, bent)

curve to the horizon and the height above the level of the liquid . Resolving vertically we find that the weight of the liquid raised above the level must be equal to T(See also:

• SIN
• SIN (O. Eng. syn: a common Teutonic word, cf. Dutch zonde, Ger. Siinde)

sin 02—sin B1), and this is therefore equal to the See also:

• AREA

area P1P2A2A1 multiplied by gp . The form of the capillary surface is identical with that of the " elastic curve," or the curve formed by a See also:

• UNIFORM

uniform See also:

• SPRING (from " to spring," " to leap or jump up," " burst out," O. Eng., springau, a common Teut. word, cf. Ger. springer, possibly allied to Gr. QnrFpxecOac, to move rapidly)

spring originally straight, when its ends are acted on by equal and rz opposite forces applied either to the ends themselves or to solid pieces attached to them . Drawings of the different forms of the curve may be found in See also:

• THOMSON, JAMES (1822-1892)
• THOMSON, JAMES (1834-1882)
• THOMSON, JAMES (r700-1748)
• THOMSON, JOHN (1778-184o)
• THOMSON, JOSEPH (1858-1895)
• THOMSON, SIR CHARLES WYVILLE (1830-1882)
• THOMSON, THOMAS (1773-1852)
• THOMSON, WILLIAM (1819-189o)

Thomson and See also:

• TAIT, ARCHIBALD CAMPBELL (1811-1882)
• TAIT, ARTHUR FITZWILLIAM (1819-1905)
• TAIT, PETER GUTHRIE (1831-1901)

Tait's Natural See also:

• PHILOSOPHY (Gr. gthos, fond of, and vo4 (a, wisdom)

Philosophy, vol. i. p . 455 . We shall next consider the rise of a liquid between two plates of different materials for which the angles of contact are a1 and See also:

• A2(z)

a2, the distance between the plates being a, a small quantity . Since the plates are very near one another we may use the following equation of the surface as an approximation: y=hi+Ax+Bx2, h2=h1+Aa+Ba2, whence cot a1= —A, cot a2=A+2Ba T(cos al+cos a2) =pga(h1+ZAa+IBa2), whence we obtain hl=pga(cos a1+cosa2)+6(2 cot al—cot a2) h2 =-- (cos al+cosa2)+6(2 cot a2—cot al) . Let X be the force which must be applied in a horizontal direction to either plate to keep it from approaching the other, then the forces acting on the first plate are T+X in the negative direction, and T sin a1+l2gph12 in the positive direction . Hence X = zgphl2—T(1—sin al) . For the second plate X=Igph22—T(I —sin a2) . Hence X=1-,gp(h12+h22)—T{I—2(sin a1+sin a2)}, or, substituting the values of h1 and h2, z X = I (cos aj+cos a2)2 2 pga2 —T{I—*(sin al+sina2)—1 12(cosal+cosa2)(cotal+cota2)}, the remaining terms being negligible when a is small . The force, therefore, with which the two plates are drawn together consists first of a positive part, or in other words an attraction, varying inversely as the square of the distance, and second, of a negative part of repulsion See also:

• INDEPENDENT

independent of the distance .

Hence in all cases except that in which the angles al and a2 are supplementary to each other, the force is attractive when a is small enough, but when cos al and cos a2 are of different signs, as when the liquid is raised by one plate, and depressed by the other, the first See also:

• TERM

term may be so small that the repulsion indicated by the second term comes into See also:

• PLAY

play . The fact that a pair of plates which repel one another at a certain distance may attract one another at a smaller distance was deduced by Laplace from theory, and verified by the observations of the See also:

• ABBE (Serbo-Croatian Rai))

abbe Haiiy . A Drop between Two Plates.—If a small quantity of a liquid which wets glass be introduced between two glass plates slightly inclined to each other, it will run towards that part where the glass plates are nearest together . When the liquid is in See also:

• EQUILIBRIUM (from the Lat. aequus, equal, and libra, a balance)

equilibrium it forms a thin film, the See also:

• OUTER

outer edge of which is all of the same thickness . If d is the distance between the plates at the edge of the film and II the atmospheric pressure, the pressure of the liquid in the film is II — 2T r s a, and if A is the area of the film between the plates and B its circumference, the plates will be pressed together with a force 2AT cos a+BT sin a, d and this, whether the See also:

• ATMOSPHERE (Gr. fir µ6r, vapour; orkaipa, a sphere)

atmosphere exerts any pressure or not . The force thus produced by the introduction of a drop of water between two plates is enormous, and is often sufficient to See also:

• PRESS (through Fr. presse from Lat. pressare, frequentative of premere, to crush, squeeze, press)

press certain parts of the plates together so powerfully as to bruise them or break them . When two blocks of See also:

• ICE (a word common to Teutonic languages; cf. Ger. Eis)

ice are placed loosely together so that the superfluous water which melts from them may drain away, the remaining water draws the blocks together with a force sufficient to cause the blocks to adhere by the process called Regelation . [An effect of an opposite See also:

• CHARACTER (Gr. xapareri7p, from xap&crew, to scratch)

character may be observed when the fluid is mercury in place of water . When two pieces of flat glass are pressed together under mercury with moderate force they cohere, the mercury leaving the narrow crevasses, even although the alternative is a vacuum . The course of events is more easily followed if one of the pieces of glass constitutes the bottom, or a See also:

• SIDE
• SIDE (mod. Eski Adalia)

side, of the vessel containing the mercury.] In many experiments bodies are floated on the surface of water in order that they may be free to move under the See also:

• ACTION

action of slight horizontal forces . Thus See also:

• SIR

Sir See also:

• ISAAC (Hebrew for " he laughs," on explanatory references to the name, see ABRAHAM)

Isaac See also:

• NEWTON
• NEWTON, ALFRED (1829–1907)
• NEWTON, JOHN (1725-1807)
• NEWTON, JOHN (1823-1895)
• NEWTON, SIR CHARLES THOMAS (1816-1894)
• NEWTON, SIR ISAAC (1642-1727)

Newton placed a magnet in a floating vessel and a piece of See also:

• IRON [symbol Fe, atomic weight 55.85 (0=16)]

iron in another in order to observe their mutual action, and A . M .

See also:

• AMPERE, ANDRE MARIE (1775–1836)
• AMPERE, JEAN JACQUES (1800-1864)

Ampere floated a voltaic See also:

• BATTERY (Fr. batterie, from battles, to beat)

battery with a coil of See also:

• WIRE (A.S. wir, a wire; cf. Swed. vire, to twist, M.H.G. wiere, a gold ornament, Lat. viriae, armlets, ultimately from the root wi, to twist, bind)

wire in its See also:

• CIRCUIT (Lat. circuitus, from circum, round, and ire, to go)

circuit in order to observe the effects of the See also:

• EARTH
• EARTH (a word common to Teutonic languages, cf. Ger. Erde, Dutch aarde, Swed. and Dan. jord; outside Teutonic it appears only in the Gr. 'pq'e, on the ground; it has been connected by some etymologists with the Aryan root ar-, to plough, which is seen in
• EARTH, FIGURE OF
• EARTH, FIGURE OF THE

earth's See also:

• MAGNETISM
• MAGNETISM, TERRESTRIAL

magnetism on the electric circuit . When such floating bodies come near the edge of the vessel they are drawn up to it, and are See also:

• APT

apt to stick fast to it . There are two ways of avoiding this inconvenience . One is to grease the See also:

• FLOAT (in O. Eng. floc and flota, in the verbal form f eotan; the Teutonic root is flut-, another form of flu-, seen in " flow," cf. " fleet "; the root is seen in Gr. a-M e, to sail, Lat. pluere, to rain; the Lat, fluere and fluctus, wave, is not connect

float round its water-line so that the water is depressed round it . This, however, often produces a worse disturbing effect, because a thin film of grease spreads over the water and increases its surface-viscosity . The other method is to fill the vessel with water till the level of the water stands a little higher than the rim of the vessel . The float will then be repelled from the edge of the vessel . Such floats, however, should always be made so that the section taken at the level of the water is as small as possible . [The Size of Drops.—The relation between the See also:

• DIAMETER (from the Cr. &a, through, µErpov, measure)

diameter of a tube and the weight of the drop which it delivers appears to have been first investigated by See also:

• THOMAS
• THOMAS (c. 1654-1720)
• THOMAS (d. 110o)
• THOMAS, ARTHUR GORING (1850-1892)
• THOMAS, CHARLES LOUIS AMBROISE (1811-1896)
• THOMAS, GEORGE (c. 1756-1802)
• THOMAS, GEORGE HENRY (1816-187o)
• THOMAS, ISAIAH (1749-1831)
• THOMAS, PIERRE (1634-1698)
• THOMAS, SIDNEY GILCHRIST (1850-1885)
• THOMAS, ST
• THOMAS, THEODORE (1835-1905)
• THOMAS, WILLIAM (d. 1554)

Thomas See also:

• TATE, JAMES (1771-1843)
• TATE, NAHUM (1652-1715)
• TATE, RALPH (1840-1901)
• TATE, SIR HENRY, BART

Tate (Phil . Mug. vol. See also:

• XXVII

xxvii. p . 176, 1864), whose experiments led him to the conclusion that " other things being the same, the weight of a drop of liquid is proportional to the diameter of the tube in which it is formed." Sufficient See also:

• TIME (0. Eng. Lima, cf. Icel. timi, Swed. timme, hour, Dan. time; from the root also seen in " tide," properly the time of between the flow and ebb of the sea, cf. O. Eng. getidan, to happen, " even-tide," &c.; it is not directly related to Lat. tempus)
• TIME, MEASUREMENT OF
• TIME, STANDARD

time must of course be allowed for the formation of the drops; otherwise no simple results can be expected., In Tate's experiments the See also:

• PERIOD
• PERIOD (Gr. irepioSos, a going or way round, circuit, aepi, round, and &Sis, way, road)

period was never less than 4o seconds . The magnitude of a drop delivered from a tube, even when the formation up to the phase of instability is infinitely slow, cannot be calculated a priori .

The weight is sometimes equated to the product of the capillary tension (T) and the circumference of the tube (2ra), but with little See also:

• JUSTIFICATION

justification . Even if the tension at the circumference of the tube acted vertically, and the whole of the liquid below this level passed into the drop, the calculation would still be vitiated by the See also:

• ASSUMPTION, FEAST OF

assumption that the internal pressure at the level in question is atmospheric . It would be necessary to consider the curvatures of the fluid surface at the edge of See also:

• ATTACHMENT

attachment . If the surface could be treated as a cylindrical prolongation of the tube (radius a), the pressure would be T/a, and the resulting force acting downwards upon the drop would amount to one-half (See also:

• RAT (a word common to Teut. and Rom. languages; probably first adopted in Teut.; the ultimate origin is not known; Skeat suggests the root rad-, to scratch; cf. Ger. Ratte, Dan. rotte, Fr. rat, &c.)

raT) of the See also:

• DIRECT

direct upward pull of the tension along the circumference . At this a, See also:

• RATE

rate the drop would be but one-half of that above reckoned . But the truth is that a See also:

• COMPLETE

complete See also:

• SOLUTION (from Lat. solvere, to loosen, dissolve)

solution of the statical problem for all forms up to that at which instability sets in, would not suffice for the present purpose . The detachment of the drop is a dynamical effect, and it is influenced by See also:

• COLLATERAL (from Med. Lat. collateralis,—cum, with, and latus, lateris, side,—side by side, hence parallel or additional)

collateral circumstances . For example, the See also:

• BORE

bore of the tube is no longer a See also:

• MATTER

matter of indifference, even though the attachment of the drop occurs entirely at the outer edge . It appears that when the See also:

• EXTERNAL

external diameter exceeds a certain value, the weight of a drop of water is sensibly different in the two extreme cases of a very small and of a very large bore . But although a complete solution of the dynamical problem is impracticable, much interesting See also:

• INFORMATION (from Lat. informare, to give shape or form to, to represent, describe)

information may be obtained from the principle of dynamical similarity . The argument has already been applied by See also:

• DUPRE, JULES (1812-1889)

Dupre (Theorie mecanique de la chaleur, See also:

• PARIS
• PARIS (also called ALEXANDROS)
• PARIS, ALEXIS PAULIN (1800-x881)
• PARIS, BRUNO PAULIN GASTON (1839-1903)
• PARIS, FRANCOIS DE (169o-1727)
• PARIS, LOUIS PHILIPPE ALBERT
• PARIS, TREATIES OF (1814-1815)

Paris, 1869, p . 328), but his presentation of it is rather obscure .

We will assume that when, as in most cases, viscosity maybe neglected, the See also:

• MASS (O.E. maesse; Fr. messe; Ger. Messe; Ital. messa; from eccl. Lat. missa)
• MASS, IN

mass (M) of a drop depends only upon the density (o•), the capillary tension (T), the See also:

• ACCELERATION (from Lat. accelerare, to hasten, celer, quick)

acceleration of gravity (g), and the linear See also:

• DIMENSION (from Lat. dimensio, a measuring)

dimension of the tube (a) . In order to justify this assumption, the formation of the drop must be sufficiently slow, and certain restrictions must be imposed upon the shape of the tube . For example, in the case of water delivered from a glass tube, which is cut off square and held vertically, a will be the external radius; and it will be necessary to suppose that the ratio of the internal radius to a is constant, the cases of a ratio infinitely small, or infinitely near unity, being included . But if the fluid be mercury, the flat end of the tube remains unwetted, and the formation of the drop depends upon the internal diameter only . The " dimensions " of the quantities on which M depends are: a.= (Mass)' (Length)-2, T = (See also:

• FARCE

Farce)' (Length)-' = (Mass)' (Time) -2, g = Acceleration = (Length)' (Time)-2, of which M, a mass, is to be expressed as a See also:

• FUNCTION

function . If we assume M T=.gY.a .au, we have, considering in turn length, time and mass, y-3Z+u=0, 2x+2y=o, x+Z= I ; so that y= -x, z =I - x, u =3-2x . Ma T¢( T )x_1 g gva2 Since x is undetermined, all that we can conclude is that M is of the form / \ M= See also:

• GAS

gas( a) (I) where F denotes an arbitrary function . Dynamical similarity requires that T/gva2 be constant; or, if g be supposed to be so, that ¢2 varies as T/v . If this See also:

• CONDITION (Lat. condicio, from condicere, to agree upon, arrange; not connected with conditio, from condere, conditum, to put together)

condition be satisfied, the mass (or weight) of the drop is proportional to T and to a . If Tate's See also:

• LAW
• LAW (0. Eng. lagu, M. Eng. lawe; from an old Teutonic root lag, " lie," what lies fixed or evenly; cf. Lat. lex, Fr. loi)
• LAW, JOHN (1671—1729)
• LAW, WILLIAM (1686-1761)

law be true, that ceteris paribus M varies as a, it follows from (1) that F is constant . For all fluids and for all similar tubes similarly wetted, the weight of a drop would then be proportional not only to the diameter of the tube, but also to the superficial tension, and it would be independent of the density . Careful observations with See also:

• SPECIAL

special precautions to ensure the cleanliness of the water have shown that over a considerable range, the departure from Tate's law is not great .

The results give material for the determination of the function F in (r) . T/9va2 gM/Ta 2.58 4'13 1.16 3'97 0.708 3.80 0'441 3'73 0.277 3'78 0.220 3.90 o 169 4.06 In the preceding table, applicable to thin-walled tubes, the first column gives the values of T/gva2, and the second column those of gM/Ta, all the quantities concerned being in C.G.S. measure, or other consistent See also:

• SYSTEM

system . From this the weight of a drop of any liquid of which the density and surface tension are known, can be calculated . For many purposes it may suffice to treat F as a constant, say 3.8 . The See also:

• FORMULA
• FORMULA (Lat. diminutive of forma, shape, pattern, &c., especially used of rules of judicial procedure)

formula for the weight of a drop is then simply Mg=3.8Ta, (2) in which 3.8 replaces the 2rr of the faulty theory alluded to earlier (see Rayleigh, Phil . See also:

• HAG

Hag., Oct . 1899).] Phenomena arising from the Variation of the Surface-tension.—Pure water has a higher surface-tension than that of any other substance liquid at ordinary temperatures except mercury . Hence any other liquid if mixed with water 'diminishes its surface-tension . For example, if a drop of See also:

• ALCOHOL

alcohol be placed on the surface of water, the surface-tension will be diminished from 8o, the value for pure water, to 25, the value for pure alcohol . The surface of the liquid will therefore no longer be in equilibrium, and a current will be formed at and near the surface from the alcohol to the surrounding water, and this current will go on as long as there is more alcohol at one part of the surface than at another . If the vessel is deep, these currents will be balanced by See also:

• COUNTER

counter currents below them, but if the See also:

• DEPTH

depth of the water is only two or three millimetres, the surface-current will sweep away the whole of the water, leaving a dry spot where the alcohol was dropped in . This phenomenon was first described and explained by See also:

• JAMES
• JAMES (Gr. 'IlrKw,l3or, the Heb. Ya`akob or Jacob)
• JAMES (JAMES FRANCIS EDWARD STUART) (1688-1766)
• JAMES, 2ND EARL OF DOUGLAS AND MAR(c. 1358–1388)
• JAMES, DAVID (1839-1893)
• JAMES, EPISTLE OF
• JAMES, GEORGE PAYNE RAINSFOP
• JAMES, HENRY (1843— )
• JAMES, JOHN ANGELL (1785-1859)
• JAMES, THOMAS (c. 1573–1629)
• JAMES, WILLIAM (1842–1910)
• JAMES, WILLIAM (d. 1827)

James Thomson, who also explained a phenomenon,the converse of this, called the " tears of strong See also:

• WINE (Lat. vinum, Gr. oivos)

wine." If a wine-glass be half-filled with See also:

• PORT

port wine the liquid rises a little up the side of the glass as other liquids do .

The wine, however, contains alcohol and water, both of which evaporate, but the alcohol faster than the water, so that the superficial layer becomes more watery . In the See also:

• MIDDLE

middle of the vessel the superficial layer recovers its strength by See also:

• DIFFUSION (from the Lat. diffundere; dis-, asunder, and fundere, to pour out)

diffusion from below, but the film adhering to the side of the glass becomes more watery, and therefore has a higher surface-tension than the surface of the stronger wine . It therefore creeps up the side of the glass dragging the strong wine after it, and this goes on till the quantity of fluid dragged up collects into a drop and runs down the side of the glass . The See also:

• MOTION (Lat. motio, from movere, to move)
• MOTION, LAWS OF

motion of small pieces of camphor floating on water arises from the gradual solution of the camphor . If this takes place more rapidly on one side of the piece of camphor than on the other side, the surface-tension becomes weaker where there is most camphor in solution, and the lump, being pulled unequally by the surface-tensions, moves off in the direction of the strongest tension, namely, towards the side on which least camphor is dissolved . If a drop of See also:

• ETHER, (C2H5)2O

ether is held near the surface of water the vapour of ether condenses on the surface of the water, and surface-currents are formed flowing in every direction away from under the drop of ether . If we place a small floating See also:

• BODY

body in a shallow vessel of water and wet one side of it with alcohol or ether, it will move off with great velocity and skim about on the surface of the water, the part wet with alcohol being always the stern . The surface-tension of mercury is greatly altered by slight changes in the See also:

• STATE
• STATE, GREAT OFFICERS OF

state of the surface . The surface-tension of pure mercury is so great that it is very difficult to keep it clean, for every See also:

• KIND (O. E. ge-cynde, from the same root as is seen in " kin," supra)

kind of oil or grease spreads over it at once . But the most remarkable effects of change of surface-tension are those produced by what is called the electric polarization of the surface . The tension of the surface of contact of mercury and dilute sulphuric acid depends on the electromotive force acting between the mercury and the acid . If the electromotive force is from the acid to the mercury the surface-tension increases; if it is from the mercury to the acid, it diminishes .

See also:

• FARADAY, MICHAEL (1791-1867)

Faraday observed that a large drop of mercury, resting on the flat bottom of a vessel containing dilute acid, changes its form in a remarkable way when connected with one of the electrodes of a battery, the other electrode being placed in the acid . When the mercury Accordingly is made positive it becomes dull and spreads itself out; when it is made negative it gathers itself together and becomes See also:

• BRIGHT, JOHN (1811-188g)
• BRIGHT, SIR CHARLES TILSTON

bright again . G . Lippmann, who has made a careful investigation of the subject, finds that exceedingly small See also:

• VARIATIONS
• VARIATIONS, CALCULUS
• VARIATIONS, CALCULUS OF

variations of the electromotive force produce sensible changes in the surface-tension . The effect of one of a See also:

• DANIELL, JOHN FREDERIC (1790-1845)
• DANIELL, THOMAS (1749–1840)

Daniell's See also:

• CELL (from Lat. cella, probably from an Indo-European kal —seen in Lat. celare, to hide; another suggestion connects the word with Lat. cera, wax, taking the original meaning to refer to the honeycomb)

cell is to increase the tension from 30.4 to 4o•6 . He has constructed a capillary See also:

• ELECTROMETER

electrometer by which See also:

• DIFFERENCES, CALCULUS OF (Theory of Finite Differences)

differences of electric potential less than o•or of that of a Daniell's cell can be detected by the difference of the pressure required to force the mercury to a given point of a See also:

• FINE

fine capillary tube . He has also constructed an apparatus in which this variation in the surface-tension is made to do See also:

• WORK

work and drive a See also:

• MACHINE
• MACHINE (through Fr. from Lat. form machina of Gr. µnxavil)

machine . He has also found that this action is reversible, for when the area of the surface of contact of the acid and mercury is made to increase, an electric current passes from the mercury to the acid, the amount of See also:

• ELECTRICITY

electricity which passes while the surface increases by one square centimetre being sufficient to decompose •000013 gramme of water . [The movements of camphor scrapings referred to above afford a useful test of the condition of a water surface . If the contamination exceed a certain limit, the scrapings remain quite dead . In a striking form of the experiment, the water is contained, to the depth of perhaps one See also:

• INCH (O. Eng. ynce from Lat. uncia, a twelfthpart; cf. " ounce," and see As)

inch, in a large flat dish, and the operative part of the surface is limited by a flexible hoop of thin See also:

• SHEET

sheet See also:

• BRASS
• BRASS (O. Eng. braes)

brass lying in the dish and rising above the water-level . If the hoop enclose an area of (say) one-third of the maximum, and if the water be clean, camphor fragments floating on the interior enter with vigorous movements .

A See also:

• TOUCH (derived through Fr. toucher from a common Teutonic and Indo-Germanic root, cf. " tug," " tuck," O. H. Ger. zucchen, to twitch or draw)

touch of the See also:

• FINGER

finger will then often reduce them to quiet; but if the hoop be See also:

• EXPANDED

expanded, the included grease is so far attenuated as to lose its effect . Another method of removing grease is to immerse and remove strips of paper by which the surface available for the contamination is in effect increased . The thickness of the film of oil adequate to check the camphor movements can be determined with See also:

• FAIR

fair accuracy by depositing a weighed amount of oil (such as •8 mg.) upon the surface of water in a large See also:

• BATH
• BATH, THOMAS THYNNE
• BATH, WILLIAM

bath . Calculated as if the density were the same as in a normal state, the thickness of the film is found to be about two millionths of a millimetre . Small as is the above amount of oil, the camphor test is a comparatively coarse one . Conditions of a contaminated surface may easily be distinguished, upon all of which camphor fragments spin vigorously . Thus, a shallow See also:

• TIN (Lat.- stannum, whence the chemical symbol " Sn "; atomic weight =117.6, 0=16)

tin vessel, such as the lid of a See also:

• BISCUIT (pronounced according to the old spelling " bisket," a Fr. form from Lat. bis, twice, and coctum, cooked, in reference to the original method of preparation; cf. Ital. biscotto, Sp. bizcocho, &c.)

biscuit See also:

• BOX
• BOX (Gr. irl or, Lat. buxus, box-wood; cf. 2r6Eir, a pyx)

box, may be levelled and filled with tap-water through a See also:

• RUBBER, INDIARUBBER

rubber See also:

• HOSE (a word common to many Teutonic languages; cf. Dutch, hoes, stocking, Ger. Hose, breeches, tights; the ultimate origin is unknown)

hose . Upon the surface of the water a little See also:

• SULPHUR

sulphur is dusted . An application of the finger for 20 or 30 seconds to the under surface of the vessel will then generate enough See also:

• HEAT (0. E. haktu, which like " hot," Old Eng. hat, is from the Teutonic type haita, hit, to be hot; cf. Ger. hitze, heiss; Dutch, hitte, heet, &c.)

heat to lower appreciably the surface-tension, as is evidenced by the opening out of the dust and the formation of a bare spot perhaps IZ in. in diameter . When, however, the surface is but very slightly greased, a spot can no longer be cleared by the warmth of the finger, or even of a spirit See also:

• LAMP (from Gr. Xag ras, a torch, Xaµaav, to shine)

lamp, held underneath . And yet the greasing may be so slight that camphor fragments move with apparently unabated vigour . The varying degrees of contamination to which a water surface is subject are the cause of many curious phenomena .

Among these is the superficial viscosity of See also:

• PLATEAU (a French term, older platel, for a flat piece of wood, metal, &c., from plat, flat)
• PLATEAU, JOSEPH ANTOINE FERDINAND (1801-1883)

Plateau . In his experiments a long See also:

• COMPASS (Fr. corn pas, ultimately from Lat. cum, with, and passus, step)

compass See also:

• NEEDLE (O. Eng. ncedl; the word appears in various forms in Teutonic languages, Ger. Nadel, Dutch naal, the root being ne-, to sew, cf. Ger. nahen, and probably Lat. nere, to spin, Gr. vi)ats, spinning)

needle is mounted so as to See also:

• SWING, DAVID (1830-1894)

swing in the surface of the liquid under investigation . The cases of ordinary clean water and alcohol are strongly contrasted, the motion of the needle upon the former being comparatively sluggish . Moreover, a different behaviour is observed when the surfaces are slightly dusted over . In the case of water the whole of the surface in front of the needle moves with it, while on the other See also:

• HAND
• HAND (a word common to Teutonic languages; cf. Ger. Hand, Goth. handus)
• HAND, FERDINAND GOTTHELF (1786-185r)

hand the dust floating on alcohol is scarcely disturbed until the needle actually strikes it . Plateau attributed these differences to a special quality of the liquids, named by him " superficial viscosity." It has been proved, however, that the question is one of contamination, and that a water surface may be prepared so as to behave in the same manner as alcohol . Another consequence of the tendency of a moderate contamination to distribute itself uniformly is the calming effectof oil, investigated by B . See also:

• FRANKLIN
• FRANKLIN, BENJAMIN (1706-1790)
• FRANKLIN, SIR JOHN (1786-1847)
• FRANKLIN, WILLIAM BUEL (1823-1903)

Franklin . On pure water the See also:

• PROPAGATION

propagation of waves would be attended by temporary extensions and contractions of the surface, but these, as was shown by 0 . See also:

• REYNOLDS, JOHN FULTON (182o-1863)
• REYNOLDS, SIR JOSHUA (1723-1792)
• REYNOLDS, WALTER (d. 1327)

Reynolds, are resisted when the surface is contaminated . Indeed the possibility of the continued existence of films, such as constitute foam, depends upon the properties now under See also:

• CONSIDERATION (from Lat. considerare, to look at closely, examine, generally taken to be from con-, and the base seen in sidus, sideris, a star, the word being supposed to be originally an astrological or astronomical term)

consideration . If, as is sometimes stated, the tension of a vertical film were absolutely the same throughout, the middle parts would of See also:

• NECESSITY (Lat. necessitas)

necessity fall with the acceleration of gravity .

In reality, the tension adjusts itself automatically to the weight to be supported at the various levels . Although throughout a certain range the surface-tension varies rapidly with the degree of contamination, it is remarkable that, as was first fully indicated by See also:

• MISS

Miss Pockels,the earlier stages of contamination have little or no effect upon surface-tension . See also:

• LORD
• LORD (O. Eng. hldford, i.e. hldfweard, the warder or keeper of bread, hlIf, loaf; the word is not represented in any other Teutonic language)
• LORD, JOHN (1810-1894)

Lord Rayleigh has shown that the fall of surface-tension begins when the quantity of oil is about the half of that required to stop the camphor movements, and he suggests that this See also:

• STAGE (Fr. 6/age; from Lat. stare, to stand)

stage may correspond with a complete coating of the surface with a single layer of molecules.] On the Forms of Liquid Films which are Figures of Revolution.—A See also:

• SOAP

soap bubble is simply a small quantity of soap-suds spread out so as to expose a large surface to the air . The bubble, in fact, has two surfaces, an outer and an inner so spahpe-rical surface, both exposed to air . It has, therefore, a bubble. certain amount of surface-energy depending on the area of these two surfaces . Since in the case of thin films the outer and inner surfaces are approximately equal, we shall consider the area of the film as representing either of them, and shall use the See also:

• SYMBOL (Gr. o-uµ(3oXov, a sign)

symbol T to denote the energy of unit of area of the film, both surfaces being taken together . If T' is the energy of a single surface of the liquid, T the energy of the film is 2T' . When by means of a tube we See also:

• BLOW, JOHN (1648-1708)

blow air into the inside of the bubble we in-crease its volume and therefore its surface, and at the same time we do work in forcing air into it, and thus increase the energy of the bubble . That the bubble has energy may be shown by leaving the end of the tube open . The bubble will See also:

• CONTRACT
• CONTRACT (Lat. contract us, from contrahere, to draw together, to bind)

contract, forcing the air out, and the current of air blown through the tube may be made to deflect the flame of a See also:

• CANDLE (Lat. candela, from candere, to glow)

candle . If the bubble is in the form of a See also:

• SPHERE (Gr. vclsa?pa, a ball or globe)

sphere of radius r this material surface will have an area S = 4711'2 (I) If T be the energy corresponding to unit of area of the film the surface-energy of the whole bubble will be ST = 41rr2T (2) The increment of this energy corresponding to an increase of the radius from r to r+dr is therefore TdS=87rrTdr (3) Now this increase of energy was obtained by forcing in air at a pressure greater than the atmospheric pressure, and thus increasing the volume of the bubble . Let II be the atmospheric pressure and II+p the pressure of the air within the bubble .

The volume of the sphere is V = 3 ~rr3, (4) and the increment of volume is dV =4ar2dr (5) Now if we suppose a quantity of air already at the pressure II+p, the work done in forcing it into the bubble is pdV . Hence the equation of work and energy is p dV=Tds (6) or 4arpr2dr = 8rrdrT (q) p=2T/r (8) This, therefore, is the excess of the pressure of the air within the bubble over that of the external air, and it is due to the action of the inner and outer surfaces of the bubble . We may conceive this pressure to arise from the tendency which the bubble has to contract, or in other words from the surface-tension of the bubble . If to increase the area of the surface requires the See also:

• EXPENDITURE

expenditure or of work, the surface must resist See also:

• EXTENSION (Lat. ex, out ; tendere, to stretch)

extension, and if the bubble in contracting can do work, the surface must tend to contract . The surface must therefore See also:

• ACT (Lat. actus, actum)

act like a sheet of See also:

• INDIA
• INDIA, FRENCH

india-rubber when extended both in length and breadth, that is, it must exert surface-tension . The tension of the sheet of india-rubber, however, depends on the extent to which it is stretched, and may be different in different directions, whereas the tension of the surface of a liquid remains the same however much the film is extended, and the tension at any point is the same in all directions . The intensity of this surface-tension is measured by the stress which it exerts across a line of unit length . Let us measure it in the case of the spherical soap-bubble by considering the stress exerted by one hemisphere of the bubble on the other, across the circumference of a great circle . This stress is balanced by the pressure p acting over the area of the same great circle: it is therefore equal to irr2p . To determine the intensity of the surface-tension we have to See also:

• DIVIDE

divide this quantity by the length of the line across which it acts, which is in this case the circumference of a great circle 27rr . Dividing ar2p by this length we obtain pr as the value of the intensity of the surface-tension, and it is See also:

• PLAIN (O. Fr. plain, from Lat. plenum)

plain from equation 8 that this is equal to T . Hence the numerical value of the intensity of the surface-tension is equal to the numerical value of the surface-energy per unit of surface .

We must remember that since the film has two surfaces the surface-tension of the film is See also:

• DOUBLE
• DOUBLE (from the Mid. Eng. duble, the form which gives the present pronunciation, through the Old Er. duble, from Lat. duplus, twice as much)

double the tension of the surface of the liquid of which it is formed . To determine the relation between the surface-tension and the pressure which balances it when the form of the surface is not spherical, let us consider the following case: Let fig . 9 represent a section through the See also:

• AXIS (Lat. for " axle ")

axis Cc of a soap-bubble in the form of a figure of revolution bounded by two circular disks AB and ab, and having the See also:

• MERIDIAN
• MERIDIAN (from the Lat. meridianus, pertaining to the south or mid-day)

meridian section APa . Let PQ be animaginary sectionnormalto the axis . Let the radius of this section PR by y, and let PT, the tangent at P, make an angle,a with the axis . Let us consider the stresses which are exerted across this imaginary section by the lower part on the upper part . If the internal pressure exceeds the external pressure by p, there is in the first place a force lry2p acting upwards arising from the pressure p over the area of the section . In .the next place, there is the surface-tension acting downwards, but at an angle a with the vertical, across the circular section of the bubble itself, whose circumference is 27ry, and the downward force is therefore 2lryT cos a . Now these forces are balanced by the external force which acts on the disk ACB, which we may call F . Hence equating the forces which act on the portion included between ACB and PRQ 7ry2p -2lryT cos a = —F (9) . If we make CR=,, and suppose z to vary, the shape of the bubble of course remaining the same, the values of y and of a will change, but the other quantities will be constant: In studying these variations we may if we please take as our independent variable the length s of the meridian section AP reckoned from A . Differentiating equation 9 with respect to s we obtain, after dividing by 27r as a See also:

• COMMON

common See also:

• FACTOR (from Lat. facere, to make or do)

factor, pyTs—T cos ads+Ty sin See also:

• ADO (d. 874)

ado=o .

. . (1o) . Now ( ) . ds = sin a The radius of curvature of the meridian section is (I2)_ R,= da . The radius of curvature of a normal section of the surface at right angles to the meridian section is equal to the part of the normal cut off by the axis, which is See also:

• R2(rt-s2)

R2 = PN = y/ cos a (13) . Hence dividing equation xo by y sin a, we find p=T(I/R,+I/R2) (14) . This equation, which gives the pressure in terms of the See also:

• PRINCIPAL

principal radii of curvature, though here proved only in the case of a surface of revolution, must be true of all surfaces . For the curvature of any surface at a given point may be completely defined in terms of the positions of its principal normal sections and their radii of curvature . Before going further we may deduce from equation 9 the nature of all the figures of revolution which a liquid film can assume . Let us first determine the nature of a curve, such that if it is rolled on the axis its origin will trace out the meridian section of the bubble . Since at any instant the See also:

• ROLLING

rolling curve is rotating about the point of contact with the axis, the line drawn from this point of contact to the tracing point must be normal to the direction of motion of the tracing point . Hence if N is the point of contact, NP must be normal to the traced curve .

Also, since the axis is a tangent to the rolling curve, the See also:

• ORDINATE

ordinate PR is the perpendicular from the tracing point P on the tangent . Hence the relation between the radius vector and the perpendicular on the tangent of the rolling curve must be identical with the relation between the normal PN and the ordinate PR of the traced curve . If we write r for PN, then y =r cos a, and equation 9 becomes y2(2pr—I) p . This relation between y and r is identical with the relation between the perpendicular from the See also:

• FOCUS (Latin for " hearth " or " fireplace ")

focus of a conic section on the tangent at a given point and the See also:

• FOCAL

focal distance of that point, provided the transverse and conjugate axes of the conic are 2a and 2b respectively, where a = p,andb2=See also:

• GYP

gyp . Hence the meridian section of the film may be traced by the focus of such a conic, if the conic is made to See also:

• ROLL (O. Fr. rolle, roulle, mod. role, Lat. rotulus, dim. of rota, wheel)
• ROLL, ALFRED PHILIPPE (1846– )

roll on the axis . On the different Forms of the Meridian Line.—l . When the conic is an See also:

• ELLIPSE (adapted from Gr. EXkei 'tc, a deficiency, iXXeirely, to fall behind)

ellipse the meridian line is in the form of a See also:

• SERIES (a Latin word from serere, to join)

series of waves, and the film itself has a series of alternate swellings and contractions as represented in See also:

• FIGS

figs . 9 and Io . This form of the film is called the unduloid . ia . When the ellipse becomes a circle, the meridian line becomes a straight line parallel to the axis, and the film passes into the form of a See also:

• CYLINDER (Gr. KvAwSpos, from KvXivaety, to roll)

cylinder of revolution . lb .

As the ellipse degenerates into the straight line joining its foci, the contracted parts of the unduloid become narrower, till at last the figure becomes a series of See also:

• SPHERES, MUSIC OF THE

spheres in contact . In all these cases the internal pressure exceeds the external by 2T/a where a is the semi-transverse axis of the conic . The resultant of the internal pressure and the surface-tension is See also:

• EQUIVALENT

equivalent to a tension along the axis, and the numerical value of this tension is equal to the force due to the action of this pressure on a circle whose diameter is equal to the conjugate axis of the ellipse . 2 . When the conic is a See also:

• PARABOLA

parabola the meridian line is a See also:

• CATENARY (from Lat. eaten¢, a chain)

catenary (fig . II); the internal pressure is equal to the external pressure, and the tension along the axis is equal to 2lTm where m is the distance of the vertex from the focus . 3 . When the conic is a See also:

• HYPERBOLA

hyperbola the meridian line is in the form of a looped curve (fig . 12) . The corresponding figure of the film is called the nodoid . The resultant of the internal pressure and the surface-tension is equivalent to a pressure along the axis equal to that due to a pressure p acting on a circle whose diameter is the conjugate axis of the hyperbola . When the conjugate axis of the hyperbola is made smaller and smaller, the nodoid approximates more and more to the series of spheres touching each other along the axis .

When the conjugate axis of the hyperbola increases without limit, the loops of the nodoid are crowded on one another, and each becomes more nearly a See also:

• RING (O.E. hring; a word common to Teutonic languages; and probably cognate with the Lat. circus, Gr. KtpKOS or KpLKOS, Skt'. chakra, wheel, circle, cf. also " harangue "); in art, a band of circular shape of varying sizes, made of any material and used f

ring of circular section, without, however, ever Non-spherical soap-bubble . reaching this form . The only closed surface belonging to the series is the sphere . These figures of revolution have been studied mathematically by C . W . B . See also:

• POISSON, SIMEON DENIS (1781-1840)

Poisson,' See also:

• GOLDSCHMIDT, HERMANN (1802-1866)

Goldschmidt,2 L . L . Lindelof and F . M . N . Moigno,3 C .

E . See also:

• DELAUNAY, ELIE (1828-1891)
• DELAUNAY, LOUIS ARSENE (1826-1903)

Delaunay,' A . H . E . Lamarle,' A . See also:

• BEER

Beer,6 and V . M . A . See also:

• MANNHEIM

Mannheim,' and have been produced experimentally by Plateau 8 in the two different ways already described . The limiting conditions of the stability of these figures have been studied both mathematically and experimentally . We shall See also:

• NOTICE

notice only two of them, the cylinder and the catenoid . Stability of the Cylinder.—The cylinder is the limiting form of the unduloid when the rolling ellipse becomes a circle .

When the ellipse differs infinitely little from a circle, the equation of the meridian line becomes approximately y = a+c sin (x/a) where c is small . This is a simple See also:

• HARMONIC

harmonic See also:

• WAVE

wave-line, whose mean distance from the axis is a, whose wave-length is 27ra, and whose See also:

• AMPLITUDE (from Lat. amplus, large)

amplitude is c . The internal pressure corresponding to this unduloid is as before p = T/a . Now consider a portion of a cylindric film of length x terminated by two equal disks of radius r and containing a certain volume of air . Let one of these disks be made to approach the other by a small quantity dx . The film will swell out into the See also:

• CONVEX

convex part of an unduloid, having its largest section midway between the disks, and we have to determine whether the internal pressure will be greater or less than before . If A and C (fig . 13) are the disks, and if x the distance between the disks is equal to 7rr half the wave-length of the harmonic curve, the disks will be at the points where the curve is at its mean distance from the axis, and the pressure will therefore be T/r as before . If Al, C, are the disks, so that the distance between them is less than ar, the curve must be produced beyond the disks before it is at its mean distance from the axis . Hence in this case the mean distance is less than r, and the pressure will be greater than T/r . If, on the other hand, the disks are at A2 and C2, so that the distance between them is greater than irr, the curve will reach its mean distance from the axis before it reaches the disks . The mean distance will therefore be greater than r, and the pressure will be less than T/r .

Hence if one of the disks be made to approach the other, the internal pressure will be increased if the distance between the disks is less than half the circumference of either, and the pressure will be diminished if the distance is greater than this quantity . In the same way we may show that if the distance between the disks is increased, the pressure will be diminished or increased according as the distance is less or more than half the circumference of either . Now let us consider a cylindric film contained between two equal fixed disks A and B, and let a third disk, C, be placed midway between . Let C be slightly displaced towards A . If AC and CB are each less than half the circumference of a disk the pressure on C will increase on the side of A and diminish on the side of B . The resultant force on C will therefore tend to oppose the displacement and to bring C back to its See also:

• ORIGINAL

original Nouvelle theorie de ''action capillaire (1831) . 2 Determinatio superficiei minimae rotatione curvae data duo Punta jungentis circa datum axem ortae (See also:

• GOTTINGEN

Gottingen, 1831) . 3 Lecons de calcul See also:

• DES

des variations (Paris, 1861) . " Sur la surface de revolution dont la courbure moyenne est constante," Liouville's See also:

• JOURNAL
• JOURNAL (through Fr. from late Lat. diurnalis, daily)

Journal, vi . 6 " Theorie geometrique des rayons et centres de courbure," See also:

• BULLET (Fr. boulet, diminutive of boule, ball)

Bullet. de l'Acad. de Belgique, 1857 . 6 Tractatus de Theoria Mathematica Phaenomenorum in Liquidis actioni gravitatis detractis observatorum (See also:

• BONN

Bonn, 1857) . Journal de l'Institut, No .

1260 . 8 Statique experimentale et theorique des liquides, 1873.position . The equilibrium of C is therefore See also:

• STABLE

stable . It is easy to show that if C had been placed in any other position than the middle, its equilibrium would have been stable . Hence the film is stable as regards See also:

• LONGITUDINAL

longitudinal displacements . It is also stable as regards displacements transverse to the axis, for the film is in a state of tension, and any lateral displacement of its middle parts would produce a resultant force tending to restore the film to its original position . Hence if the length of the cylindric film is less than its circumference, it is in stable equilibrium . But if the length of the cylindric film is greater than its circumference, and if we suppose the disk C to be placed midway between A and B, and to be moved towards A, the pressure on the side next A will diminish, and that on the side next B will increase, so that the resultant force will tend to increase the displacement, and the equilibrium of the disk C is therefore unstable . Hence the equilibrium of a cylindric film whose length is greater than its circumference is unstable . Such a film, if ever so little disturbed, will begin to contract at one secton and to expand at another, till its form ceases to resemble a cylinder, if it does not break up into two parts which become ultimately portions of spheres . Instability of a See also:

• JET (Fr. jais, Ger. Gagat)

Jet of Liquid.—When a liquid flows out of a vessel through a circular opening in the bottom of the vessel, the form of the stream is at first nearly cylindrical though its diameter gradually diminishes from the orifice downwards on account of the increasing velocity of the liquid . But the liquid after it leaves the vessel is subject to no forces except gravity, the pressure of the air, and its own surface-tension .

Of these gravity has no effect on the form of the stream except in See also:

• DRAWING

drawing asunder its parts in a vertical direction, because the lowe