LUBLIN

lublin, a town of russian poland, capital of the government of the same name, 109 m. by rail s.e. of warsaw, on a small tributary of the wieprz. pop. (1873) 28,900; (1897) 50,152. it is the most important town of poland after warsaw and lodz, being one of the chief centres of the manufacture of thread-yarn, linen and hempen goods and woollen stuffs; there is also trade in grain and cattle. it has an old citadel, several palaces of polish nobles and many interesting churches, and is the headquarters of the xiv. army corps, and the see of a roman catholic bishop. the cathedral dates from the 16th century. of the former fortifications nothing remains except the four gates, one dating from 1342. lublin was in existence in the 10th century, and has a church which is said to have been built in 986. during the time the jagellon dynasty ruled over lithuania and poland it was the most important city between the vistula and the dnieper, having 40,000 inhabitants (70,000 according to other authorities) and all the trade with podolia, volhynia and red russia. indeed, the present town is surrounded with ruins, which prove that it formerly covered a much larger area. but it was frequently destroyed by the tatars (e.g. 1240) and cossacks (e.g. 1477). in 1568-1569 it was the seat of the stormy convention at which the union between poland and lithuania was decided. in 1702 another convention was held in lublin, in favour of augustus ii. and against charles xii. of sweden, who carried the town by assault and plundered it. in 1831 lublin was taken by the russians. the surrounding country is rich in reminiscences of the struggle of poland for independence. lubricants. machines consist of parts which have relative motion and generally slide and rub against each other. thus the axle of a cart or railway vehicle is pressed against a metallic bearing surface supporting the body of the vehicle, and the two opposed surfaces slide upon each other and are pressed together with great force. if the metallic surfaces be clean, the speed of rubbing high, and the force pressing the surfaces together considerable, then the latter will abrade each other, become hot and be rapidly destroyed. it is possible, however, to prevent the serious abrasion of such opposing surfaces, and largely to reduce the frictional resistance they oppose to relative motion by the use of _lubricants_ (lat. _lubricare_, _lubricus_, slippery). these substances are caused to insinuate themselves between the surfaces, and have the property of so separating them as to prevent serious abrasion. the solid and semi-solid lubricants seem to act as rollers between the surfaces, or form a film between them which itself suffers abrasion or friction. the liquid lubricants, however, maintain themselves as liquid films between the surfaces, upon which the bearing floats. the frictional resistance is then wholly in the fluid. even when lubricants are used the friction, i.e. the resistance to motion offered by the opposing surfaces, is considerable. in the article friction will be found a statement of how friction is measured and the manner in which it is expressed. the coefficient of friction is obtained by dividing the force required to cause the surfaces to slide over each other by the load pressing them together. for clean unlubricated surfaces this coefficient may be as great as 0.3, whilst for well-lubricated cylindrical bearings it may be as small as 0.0006. engineers have, therefore, paid particular attention to the design of bearings with the object of reducing the friction, and thus making use of as much as possible of the power developed by prime movers. the importance of doing this will be seen when it is remembered that the energy wasted is proportional to the coefficient of friction, and that the durability of the parts depends upon the extent to which they are separated by the lubricant and thus prevented from injuring each other. there is great diversity in the shapes of rubbing surfaces, the loads they have to carry vary widely, and the speed of rubbing ranges from less than one foot to thousands of feet per minute. there is also a large number of substances which act as lubricants, some being liquids and others soft solids. in many instruments or machines where the surfaces in contact which have to slide upon each other are only lightly pressed together, and are only occasionally given relative motion, the lubricant is only needed to prevent abrasion. microscopes and mathematical instruments are of this kind. in such cases, the lubricant which keeps the surfaces from abrading each other is a mere contamination film, either derived from the air or put on when the surfaces are finished. when such lubricating films are depended upon, the friction surfaces should be as hard as possible and, if practicable, of dissimilar metals. in the absence of a contamination film, most metals, if rubbed when in contact, will immediately adhere to each other. a large number of experiments have been made to ascertain the coefficient of friction under these imperfect conditions of lubrication. within wide limits of load, the friction is proportional to the pressure normal to the surfaces and is, therefore, approximately independent of the area of the surfaces in contact. although the static coefficient is often less than the kinetic at very low speeds, within wide limits the latter coefficient decreases with increasing speed. these laws apply to all bearings the velocity of rubbing of which is very small, or which are lubricated with solid or semi-solid materials. when the speed of rubbing is considerable and the contamination film is liable to be destroyed, resort is had to lubricants which possess the power of keeping the surfaces apart, and thereby reducing the friction. the constant application of such substances is necessary in the case of such parts of machine tools as slide rests, the surfaces of which only move relatively to each other at moderate speeds, but which have to carry heavy loads. in all ordinary cases, the coefficient of friction of flat surfaces, such as those of slide blocks or pivot bearings, is high, owing to the fact that the lubricant is not easily forced between the surfaces. in the case of cylindrical bearing surfaces, such as those of journals and spindles, owing to the fact that the radius of the bearing surface is greater than that of the journal or spindle, the lubricant, if a liquid, is easily drawn in and entirely separates the surfaces (see lubrication). fortunately, cylindrical bearings are by far the most common and important form of bearing, and they can be so lubricated that the friction coefficient is very low. the lubricant, owing to its viscosity, is forced between the surfaces and keeps them entirely apart. this property of viscosity is one of the most important possessed by liquid lubricants. some lubricants, such as the oils used for the light spindles of textile machinery, are quite thin and limpid, whilst others, suitable for steam engine cylinders and very heavy bearings, are, at ordinary temperatures, as thick as treacle or honey. generally speaking, the greater the viscosity of the lubricant the greater the load the bearing will carry, but with thick lubricants the frictional coefficient is correspondingly high. true lubricants differ from ordinary liquids of equal viscosity inasmuch as they possess the property of "oiliness." this is a property which enables them to maintain an unbroken film between surfaces when the loads are heavy. it is possessed most markedly by vegetables and animal oils and fats, and less markedly by mineral oils. in the case of mineral lubricating oils from the same source, the lower the specific gravity the greater the oiliness of the liquid, as a rule. mixtures of mineral oil with animal or vegetable oil are largely used, one class of oil supplying those qualities in which the other is deficient. thus the mineral oils, which are comparatively cheap and possess the important property of not becoming oxidized into gummy or sticky substances by the action of the air, which also are not liable to cause spontaneous ignition of cotton waste, &c., and can be manufactured of almost any desired viscosity, but which on the other hand are somewhat deficient in the property of oiliness, are mixed with animal or vegetable oils which possess the latter property in marked degree, but are liable to gum and become acid and to cause spontaneous ignition, besides being comparatively expensive and limited in quantity. oils which become acid attack the bearings chemically, and those which oxidize may become so thick that they fail to run on to the bearings properly. the following table shows that the permissible load on bearings varies greatly:-- description of bearing. load in lb. per sq. in. hard steel bearings on which the load is inter- mittent, such as the crank pins of shearing machines 3000 bronze crosshead neck journals 1200 crank pins of large slow engines 800-900 crank pins of marine engines 400-500 main crank-shaft bearings, slow marine 600 main crank-shaft bearings, fast marine 400 railway coach journals 300-400 fly-wheel shaft journals 150-200 small engine crank pins 150-200 small slide blocks, marine engines 100 stationary engine slide block 25-125 stationary engine slide block, usually 30-60 propeller thrust bearings 50-70 shafts in cast iron steps, high speed 15 _solid lubricants._--solid substances, such as graphite or plumbago, soapstone, &c., are used as lubricants when there is some objection to liquids or soft solids, but the surfaces between which they are placed should be of very hard materials. they are frequently mixed with oils or greases, the lubricating properties of which they improve. _semi-solid lubricants._--the contrast in lubricating properties between mineral and fatty oils exists also in the case of a pure mineral grease like vaseline and an animal fat such as tallow, the latter possessing in a far greater degree the property of greasiness. a large number of lubricating greases are made by incorporating or emulsifying animal and vegetable fats with soap and water; also by thickening mineral lubricating oils with soap. large quantities of these greases are used with very good results for the lubrication of railway waggon axles, and some of them are excellent lubricants for the bearings of slow moving machinery. care must be taken, however, that they do not contain excess of water and are not adulterated with such useless substances as china clay; also, that they melt as a whole, and that the oil does not run down and leave the soap. this is liable to occur with badly made greases, and hot bearings are the result. except in special cases, greases should not be used for quick-running journals, shafts or spindles, on account of the high frictional resistance which they offer to motion. in the case of fats and greases whose melting points are not much above the temperature of surrounding objects it generally happens that the lubricating films are so warmed by friction that they actually melt and act as oils. these lubricants are generally forced into the bearings by a form of syringe fitted with a spring piston, or are squeezed between the faces by means of a screw-plug. _liquid lubricants._--generally speaking, all bearings which it is necessary should run with as little friction as possible must be supplied with liquid lubricants. these may be of animal, vegetable or mineral origin. the mineral oils are mixtures of hydrocarbons of variable viscosity, flashing-point, density and oiliness. they are obtained by distillation from american, russian and other petroleums. the fixed oils obtained from animal and vegetable substances are not volatile without decomposition, and are found ready made in the tissues of animals and plants. animal oils are obtained from the adipose tissue by simple heat or by boiling with water. they are usually either colourless or yellow. the oils of plants occur usually in the seeds or fruit, and are obtained either by expression or by means of solvents such as ether or petroleum. they are of various shades of yellow and green, the green colour being due to the presence of chlorophyll. the fundamental difference between fixed oils and mineral oils exists in their behaviour towards oxygen. mineral oils at ordinary temperatures are indifferent to oxygen, but all fixed oils combine with it and thicken or gum more or less, generating heat at the same time. such oils are, therefore, dangerous if dropped upon silk, cotton or woollen waste or other combustible fibrous materials, which are thus rendered liable to spontaneous ignition. liquid lubricants are used for all high speed bearings. in some cases the rubbing surfaces work in a bath of the lubricant, which can then reach all the rubbing parts with certainty. small engines for motor cars or road waggons are often lubricated in this way. in the case of individual bearings, such as those of railway vehicles, a pad of cotton, worsted and horse hair is kept saturated with the lubricant and pressed against the under side of the journal. the journal is thus kept constantly wetted with oil, and the film is forced beneath the brass as the axle rotates. in many cases, oil-ways and grooves are cut in the bearings, and the lubricant is allowed to run by gravity into them and thus finds its way between the opposing surfaces. to secure a steady feed various contrivances are adopted, the most common being a wick of cotton or worsted used as a siphon. in cases where it is important that little if any wear should take place, the lubricant is forced by means of a pump between the friction surfaces and a constant film of oil is thereby maintained between them. for the spindles of small machines such as clocks, watches and other delicate mechanisms, which are only lubricated at long intervals and are often exposed to extremes of temperature, the lubricant must be a fluid oil as free as possible from tendency to gum or thicken by oxidation or to corrode metal, and must often have a low freezing-point. it must also possess a maximum of "oiliness." the lubricants mostly used for such purposes are obtained from porpoise or dolphin jaw oils, bean oil, hazel nut oil, neatsfoot oil, sperm oil or olive oil. these oils are exposed for some time to temperatures as low as the mechanism is required to work at, and the portion which remains fluid is separated and used. free acid should be entirely eliminated by chemical refining. a little good mineral oil may with advantage be mixed with the fatty oil. for all ordinary machinery, ranging from the light ring spindles of textile mills to the heavy shafts of large engines, mineral oils are almost universally employed, either alone or mixed with fatty oils, the general rule being to use pure mineral oils for bath, forced or circulating pump lubrication, and mixed oils for drop, siphon and other less perfect methods of lubrication. pure mineral oils of relatively low viscosity are used for high speeds and low pressures, mixed oils of greater viscosity for low speeds and high pressures. in selecting oils for low speeds and great pressures, viscosity must be the first consideration, and next to that "oiliness." if an oil of sufficiently high viscosity be used, a mineral oil may give a result as good or better than a pure fixed oil; a mixed oil may give a better result than either. if a mineral oil of sufficient viscosity be not available, then a fixed oil or fat may be expected to give the best result. in special cases, such as in the lubrication of textile machines, where the oil is liable to be splashed upon the fabric, the primary consideration is to use an oil which can be washed out without leaving a stain. pure fixed oils, or mixtures composed largely of fixed oils, are used for such purposes. in other special cases, such as marine engines working in hot places, mixtures are used of mineral oil with rape or other vegetable oil artificially thickened by blowing air through the heated oil, and known as "blown" oil or "soluble castor oil." in the lubrication of the cylinders and valves of steam, gas and oil engines, the lubricant must possess as much viscosity as possible at the working temperature, must not evaporate appreciably and must not decompose and liberate fatty acids which would corrode the metal and choke the steam passages with metallic soaps; for gas and oil engines the lubricant must be as free as possible from tendency to decompose and deposit carbon when heated. for this reason steam cylinders and valves should be lubricated with pure mineral oils of the highest viscosity, mixed with no more fixed oil than is necessary to ensure efficient lubrication. gas and oil engines also should be lubricated with pure mineral oils wherever possible. for further information on the theory and practice of lubrication and on the testing of lubricants, see _friction and lost work in machinery and mill work_, by r. h. thurston (1903); and _lubrication and lubricants_, by l. archbutt and r. m. deeley (1906). (r. m. d.) lubrication. our knowledge of the action of oils and other viscous fluids in diminishing friction and wear between solid surfaces from being purely empirical has become a connected theory, based on the known properties of matter, subjected to the definition of mathematical analysis and verified by experiment. the theory was published in 1886 (_phil. trans._, 1886, 177, pp. 157-234); but it is the purpose of this article not so much to explain its application, as to give a brief account of the introduction of the misconceptions that so long prevailed, and of the manner in which their removal led to its general acceptance. friction, or resistance to tangential shifting of matter over matter, whatever the mode and arrangement, differs greatly according to the materials, but, like all material resistance, is essentially limited. the range of the limits in available materials has a primary place in determining mechanical possibilities, and from the earliest times they have demanded the closest attention on the part of all who have to do with structures or with machines, the former being concerned to find those materials and their arrangements which possess the highest limits, and the latter the materials in which the limits are least. long before the reformation of science in the 15th and 16th centuries both these limits had formed the subject of such empirical research as disclosed numerous definite although disconnected circumstances under which they could be secured; and these, however far from the highest and lowest, satisfied the exigencies of practical mechanics at the time, thus initiating the method of extending knowledge which was to be subsequently recognized as the only basis of physical philosophy. in this purely empirical research the conclusion arrived at represented the results for the actual circumstance from which they were drawn, and thus afforded no place for theoretical discrepancies. however, in the attempts at generalization which followed the reformation of science, opportunity was afforded for such discrepancies in the mere enunciation of the circumstances in which the so-called laws of friction of motion are supposed to apply. the circumstances in which the great amount of empirical research was conducted as to the resistance between the clean, plane, smooth surfaces of rigid bodies moving over each other under pressure, invariably include the presence of air at atmospheric pressure around, and to some extent between, the surfaces; but this fact had received no notice in the enunciation of these laws, and this constitutes a theoretical departure from the conditions under which the experience had been obtained. also, the theoretical division of the law of frictional resistance into two laws--one dealing with the limit of rest, and the other asserting that the friction of motion, which is invariably less in similar circumstances than that of rest, is independent of the velocity of sliding--involves the theoretical assumption that there is no asymptotic law of diminution of the resistance, since, starting from rest, the rate of sliding increases. the theoretical substitution of ideal rigid bodies with geometrically regular surfaces, sliding in contact under pressure at the common regular surface, for the aerated surfaces in the actual circumstances, and the theoretical substitution of the absolute independence of the resistance of the rate of sliding for the limited independence in the actual circumstances, prove the general acceptance of the conceptions--(1) that matter can slide over matter under pressure at a geometrically regular surface; (2) that, however much the resistance to sliding under any particular pressure (the coefficient of friction) may depend on the physical properties of the materials, the sliding under pressure takes place at the geometrically regular surface of contact of the rigid bodies; and (3) as the consequence of (1) and (2), that whatever the effect of a lubricant, such as oil, might have, it could be a physical surface effect. thus not only did these general theoretical conceptions, resulting from the theoretical laws of friction, fail to indicate that the lubricant may diminish the resistance by the mere mechanical separation of the surfaces, but they precluded the idea that such might be the case. the result was that all subsequent attempts to reduce the empirical facts, where a lubricant was used, to such general laws as might reveal the separate functions of the complex circumstances on which lubrication depends, completely failed. thus until 1883 the science of lubrication had not advanced beyond the empirical stage. this period of stagnation was terminated by an accidental phenomenon observed by beauchamp tower, while engaged on his research on the friction of the journals of railway carriages. his observation led him to a line of experiments which proved that in these experiments the general function of the lubricant was the mechanical separation of the metal surfaces by a layer of fluid of finite thickness, thus upsetting the preconceived ideas as expressed in the laws of the friction of motion. on the publication of tower's reports (_proc. inst. m.e._, november 1883), it was recognized by several physicists (_b.a. report_, 1884, pp. 14, 625) that the evidence they contained afforded a basis for further study of the actions involved, indicating as it did the circumstances--namely, the properties of viscosity and cohesion possessed by fluids--account of which had not been taken in previous conclusions. it also became apparent that continuous or steady lubrication, such as that of tower's experiments, is only secured when the solid surfaces separated by the lubricant are so shaped that the thickness at the ingoing side is greater than that at the outgoing side. when the general equations of viscous fluids had been shown as the result of the labours of c. l. m. h. navier,[1] a. l. cauchy,[2] s. d. poisson,[3] a. j. c. barre de st venant,[4] and in 1845 of sir g. gabriel stokes,[5] to involve no other assumption than that the stresses, other than the pressure equal in all directions, are linear functions of the distortional rates of strain multiplied by a constant coefficient, it was found that the only solutions of which the equations admitted, when applied to fluids flowing between fixed boundaries, as water in a pipe, were singular solutions for steady or steady periodic motion, and that the conclusions they entailed, that the resistance would be proportional to the velocity, were for the most part directly at variance with the common experience that the resistances varied with the square of the velocity. this discrepancy was sometimes supposed to be the result of eddies in the fluid, but it was not till 1883 that it was discovered by experiments with colour bands that, in the case of geometrically similar boundaries, the existence or non-existence of such eddies depended upon a definite relation between the mean velocity (u) of the fluid, the distance between the boundaries, and the ratio of the coefficient of viscosity to the density ([mu]/[rho]), expressed by ud[rho]/[mu] = k, where k is a physical constant independent of units, which has a value between 1900 and 2000, and for parallel boundaries d is four times the area of the channel divided by the perimeter of the section (_phil. trans._, 1883, part iii. 935-982). k is thus a criterion at which the law of resistance to the mean flow changes suddenly (as u increases), from being proportional to the flow, to a law involving higher powers of the velocity at first, but as the rates increase approaching an asymptote in which the power is a little less that the square. this sudden change in the law of resistance to the flow of fluid between solid boundaries, depending as it does on a complete change in the manner of the flow--from direct parallel flow to sinuous eddying motion--serves to determine analytically the circumstances as to the velocity and the thickness of the film under which any fluid having a particular coefficient of viscosity can act the part of a lubricant. for as long as the circumstances are such that ud[rho]/[mu] is less than k, the parallel flow is held stable by the viscosity, so that only one solution is possible--that in which the resistance is the product of [mu] multiplied by the rate of distortion, as [mu](du/dy); in this case the fluid has lubricating properties. but when the circumstances are such that ud[rho]/[mu] is greater than k, other solutions become possible, and the parallel flow becomes unstable, breaks down into eddying motion, and the resistance varies as [rho]u^n, which approximates to [rho]u^(1.78) as the velocity increases; in this state the fluid has no lubricating properties. thus, within the limits of the criterion, the rate of displacement of the momentum of the fluid is insignificant as compared with the viscous resistance, and may be neglected; while outside this limit the direct effects of the eddying motion completely dominate the viscous resistance, which in its turn may be neglected. thus k is a criterion which separates the flow of fluid between solid surfaces as definitely as the flow of fluid is separated from the relative motions in elastic solids, and it is by the knowledge of the limit on which this distinction depends that the theory of viscous flow can with assurance be applied to the circumstance of lubrication. until the existence of this physical constant was discovered, any theoretical conclusions as to whether in any particular circumstances the resistance of the lubricant would follow the law of viscous flow or that of eddying motion was impossible. thus tower, being unaware of the discovery of the criterion, which was published in the same year as his reports, was thrown off the scent in his endeavour to verify the evidence he had obtained as to the finite thickness of the film by varying the velocity. he remarks in his first report that, "according to the theory of fluid motion, the resistance would be as the square of the velocity, whereas in his results it did not increase according to this law." the rational theory of lubrication does not, however, depend solely on the viscosity within the interior of fluids, but also depends on the surface action between the fluid and the solid. in many respects the surface actions, as indicated by surface tension, are still obscure, and there has been a general tendency to assume that there may be discontinuity in the velocity at the common surface. but whatever these actions may be in other respects, there is abundant evidence that there is no appreciable discontinuity in the velocity at the surfaces as long as the fluid has finite thickness. hence in the case of lubrication the velocities of the fluid at the surfaces of the solids are those of the solid. in as far as the presence of the lubricant is necessary, such properties as cause oil in spite of its surface tension to spread even against gravity over a bright metal surface, while mercury will concentrate into globules on the bright surface of iron, have an important place in securing lubrication where the action is intermittent, as in the escapement of a clock. if there is oil on the pallet, although the pressure of the tooth causes this to flow out laterally from between the surfaces, it goes back again by surface tension during the intervals; hence the importance of using fluids with low surface tension like oil, or special oils, when there is no other means of securing the presence of the lubricant. the differential equations for the equilibrium of the lubricant are what the differential equations of viscous fluid in steady motion become when subject to the conditions necessary for lubrication as already defined--(1) the velocity is below the critical value; (2) at the surfaces the velocity of the fluid is that of the solid; (3) the thickness of the film is small compared with the lateral dimensions of the surfaces and the radii of curvature of the surfaces. by the first of these conditions all the terms having [rho] as a factor may be neglected, and the equations thus become the equations of equilibrium of the fluid; as such, they are applicable to fluid whether incompressible or elastic, and however the pressure may affect the viscosity. but the analysis is greatly simplified by omitting all terms depending on compressibility and by taking [mu] constant; this may be done without loss of generality in a qualitative sense. with these limitations we have for the differential equation of the equilibrium of the lubricant:-- dp du dv dw \ 0 = -- - [mu]2u, &c., &c., 0 = -- + -- + -- | dx dx dy dz | > (1) / du dv \ | 0 = p_yx - [mu] ( -- + -- ), &c., &c. | \ dy dx / / these are subject to the boundary conditions (2) and (3). taking x as measured parallel to one of the surfaces in the direction of relative motion, y normal to the surface and z normal to the plane of xy by condition (3), we may without error disregard the effect of any curvature in the surfaces. also v is small compared with u and w, and the variations of u and w in the directions x and z are small compared with their variation in the direction y. the equations (1) reduce to dp d2u dp dp d2w du dv dw \ 0 = -- - [mu]---, 0 = --, 0 = -- - [mu]---, 0 = -- + -- + -- | dx dy2 dy dz dy2 dx dy dz | > (2) du dw | 0 = p_yx - [mu]--, 0 = p_yz - [mu]--, p_xz = 0. | dy dy / for the boundary conditions, putting f(x, z) as limiting the lateral area of the lubricant, the conditions at the surfaces may be expressed thus:-- when y = 0, u = u0, w = 0, v = 0 \ dh | when y = h, u = u1, w = 0, v1, = u1 -- + v1 > (3) dx | when f(x, z) = 0, p = p0 / then, integrating the equations (2) over y, and determining the constants by equations (3), we have, since by the second of equations (2) p is independent of y, 1 dp h - y y \ u = ----- -- (y - h)y + u0 ----- + u1 --- | 2[mu] dx h h | > (4) 1 dp | w = ----- -- (y - h)y | 2[mu] dz / then, differentiating equations (4) with respect to x and z respectively, and substituting in the 4th of equations (2), and integrating from y = 0 to y = h, so that only the values of v at the surfaces may be required, we have for the differential equation of normal pressure at any point x, z, between the boundaries:-- _ _ d / dp\ d / dp\ | dh | --- ( h3 -- ) + --- ( h3-- ) = 6[mu] | (u0 + u1) -- + 2v1 | (5) dx \ dz/ dz \ dz/ |_ dx _| again differentiating equations (4), with respect to x and z respectively, and substituting in the 5th and 6th of equations (2), and putting f_x and f_z for the intensities of the tangential stresses at the lower and upper surfaces:-- 1 h dp \ f_x = [mu](u1 + u0) --- ± --- -- | h 2 dx | > (6) h dp | f_z = ± --- -- | 2 dx / equations (5) and (6) are the general equations for the stresses at the boundaries at x, z, when h is a continuous function of x and z, [mu] and [rho] being constant. for the integration of equations (6) to get the resultant stresses and moments on the solid boundaries, so as to obtain the conditions of their equilibrium, it is necessary to know how x and z at any point on the boundary enter into h, as well as the equation f(x, z) = 0, which determines the limits of the lubricating film. if y, the normal to one of the surfaces, has not the same direction for all points of this surface, in other words, if the surface is not plane, x and z become curvilinear co-ordinates, at all points perpendicular to y. since, for lubrication, one of the surfaces must be plane, cylindrical, or a surface of revolution, we may put x = r[theta], y = r - r, and z perpendicular to the plane of motion. then, if the data are sufficient, the resultant stresses and moments between the surfaces are obtained by integrating the intensity of the stress and moments of intensity of stress over the surface. this, however, is not the usual problem that arises. what is generally wanted is to find the thickness of the film where least (h0) and its angular position with respect to direction of load, to resist a definite load with a particular surface velocity. if the surfaces are plane, the general solution involves only one arbitrary constant, the least thickness (h0); since in any particular case the variation of h with x is necessarily fixed, as in this case lubrication affords no automatic adjustment of this slope. when both surfaces are curved in the plane of motion there are at least two arbitrary constants, h0, and [phi] the angular position of h0 with respect to direction of load; while if the surfaces are both curved in a plane perpendicular to the direction of motion as well as in the plane of motion, there are three arbitrary constants, h0, [phi]0, z0. the only constraint necessary is to prevent rotation in the plane of motion of one of the surfaces, leaving this surface free to move in any direction and to adjust its position so as to be in equilibrium under the load. the integrations necessary for the solutions of these problems are practicable--complete or approximate--and have been effected for circumstances which include the chief cases of practical lubrication, the results having been verified by reference to tower's experiments. in this way the verified theory is available for guidance outside the limits of experience as well as for determining the limiting conditions. but it is necessary to take into account certain subsidiary theories. these limits depend on the coefficient of viscosity, which diminishes as the temperature increases. the total work in overcoming the resistance is spent in generating heat in the lubricant, the volume of which is very small. were it not for the escape of heat by conduction through the lubricant and the metal, lubrication would be impossible. hence a knowledge of the empirical law of the variation of the viscosity of the lubricant with temperature, the coefficients of conduction of heat in the lubricant and in the metal, and the application of the theory of the flow of heat in the particular circumstances, are necessary adjuncts to the theory of lubrication for determining the limits of lubrication. nor is this all, for the shapes of the solid surfaces vary with the pressure, and more particularly with the temperature. the theory of lubrication has been applied to the explanation of the slipperiness of ice (_mem. manchester lit. and phil. soc._, 1899). (o. r.) footnotes: [1] _mem. de l'acad._ (1826), 6, p. 389. [2] _mem. des sav. etrang._ l. 40. [3] _mem. de l'acad._ (1831), 10, p. 345. [4] _b.a. report_ (1846). [5] _cambridge phil. trans._ (1845 and 1857). lucan [marcus annaeus lucanus], (a.d. 39-65), roman poet of the silver age, grandson of the rhetorician seneca and nephew of the philosopher, was born at corduba. his mother was acilia; his father, marcus annaeus mela, had amassed great wealth as imperial procurator for the provinces. from a memoir which is generally attributed to suetonius we learn that lucan was taken to rome at the age of eight months and displayed remarkable precocity. one of his instructors was the stoic philosopher, cornutus, the friend and teacher of persius. he was studying at athens when nero recalled him to rome and made him quaestor. these friendly relations did not last long. lucan is said to have defeated nero in a public poetical contest; nero forbade him to recite in public, and the poet's indignation made him an accomplice in the conspiracy of piso. upon the discovery of the plot he is said to have been tempted by the hope of pardon to denounce his own mother. failing to obtain a reprieve, he caused his veins to be opened, and expired repeating a passage from one of his poems descriptive of the death of a wounded soldier. his father was involved in the proscription, his mother escaped, and his widow polla argentaria survived to receive the homage of statius under domitian. the birthday of lucan was kept as a festival after his death, and a poem addressed to his widow upon one of these occasions and containing information on the poet's work and career is still extant (statius's _silvae_, ii. 7, entitled _genethliacon lucani_). besides his principal performance, lucan's works included poems on the ransom of hector, the nether world, the fate of orpheus, a eulogy of nero, the burning of rome, and one in honour of his wife (all mentioned by statius), letters, epigrams, an unfinished tragedy on the subject of medea and numerous miscellaneous pieces. his minor works have perished except for a few fragments, but all that the author wrote of the _pharsalia_ has come down to us. it would probably have concluded with the battle of philippi, but breaks off abruptly as caesar is about to plunge into the harbour of alexandria. the _pharsalia_ opens with a panegyric of nero, sketches the causes of the war and the characters of caesar and pompey, the crossing of the rubicon by caesar, the flight of the tribunes to his camp, and the panic and confusion in rome, which pompey has abandoned. the second book describes the visit of brutus to cato, who is persuaded to join the side of the senate, and his marriage a second time to his former wife marcia, ahenobarbus's capitulation at corfinium and the retirement of pompey to greece. in the third book caesar, after settling affairs in rome, crosses the alps for spain. massilia is besieged and falls. the fourth book describes the victories of caesar in spain over afranius and petreius, and the defeat of curio by juba in africa. in the fifth caesar and antony land in greece, and pompey's wife cornelia is placed in security at lesbos. the sixth book describes the repulses of caesar round dyrrhachium, the seventh the defeat of pompey at pharsalia, the eighth his flight and assassination in egypt, the ninth the operations of cato in africa and his march through the desert, and the landing of caesar in egypt, the tenth the opening incidents of the alexandrian war. the incompleteness of the work should not be left out of account in the estimate of its merits, for, with two capital exceptions, the faults of the _pharsalia_ are such as revision might have mitigated or rendered. no such pains, certainly, could have amended the deficiency of unity of action, or supplied the want of a legitimate protagonist. the _pharsalia_ is not true to history, but it cannot shake off its shackles, and is rather a metrical chronicle than a true epic. if it had been completed according to the author's design, pompey, cato and brutus must have successively enacted the part of nominal hero, while the real hero is the arch-enemy of liberty and lucan, caesar. yet these defects, though glaring, are not fatal or peculiar to lucan. the false taste, the strained rhetoric, the ostentatious erudition, the tedious harangues and far-fetched or commonplace reflections so frequent in this singularly unequal poem, are faults much more irritating, but they are also faults capable of amendment, which the writer might not improbably have removed. great allowance should also be made in the case of one who is emulating predecessors who have already carried art to its last perfection. lucan's temper could never have brooked mere imitation; his versification, no less than his subject, is entirely his own; he avoids the appearance of outward resemblance to his great predecessor with a persistency which can only have resulted from deliberate purpose, but he is largely influenced by the declamatory school of his grandfather and uncle. hence his partiality for finished antithesis, contrasting strongly with his generally breathless style and turbid diction. quintilian sums up both aspects of his genius with pregnant brevity, "ardens et concitatus et sententiis clarissimus," adding with equal justice, "magis oratoribus quam poetis annumerandus." lucan's oratory, however, frequently approaches the regions of poetry, e.g. the apotheosis of pompey at the beginning of the ninth book, and the passage in the same book where cato, in the truest spirit of the stoic philosophy, refuses to consult the oracle of jupiter ammon. though in many cases lucan's rhetoric is frigid, hyperbolical, and out of keeping with the character of the speaker, yet his theme has a genuine hold upon him; in the age of nero he celebrates the republic as a poet with the same energy with which in the age of cicero he might have defended it as an orator. but for him it might almost have been said that the roman republic never inspired the roman muse. lucan never speaks of himself, but his epic speaks for him. he must have been endowed with no common ambition, industry and self-reliance, an enthusiastic though narrow and aristocratic patriotism, and a faculty for appreciating magnanimity in others. but the only personal trait positively known to us is his conjugal affection, a characteristic of seneca also. lucan, together with statius, was preferred even to virgil in the middle ages. so late as 1493 his commentator sulpitius writes: "magnus profecto est maro, magnus lucanus; adeoque prope par, ut quis sit major possis ambigere." shelley and southey, in the first transport of admiration, thought lucan superior to virgil; pope, with more judgment, says that the fire which burns in virgil with an equable glow breaks forth in lucan with sudden, brief and interrupted flashes. of late, notwithstanding the enthusiasm of isolated admirers, lucan has been unduly neglected, but he has exercised an important influence upon one great department of modern literature by his effect upon corneille, and through him upon the classical french drama. authorities.--the _pharsalia_ was much read in the middle ages, and consequently it is preserved in a large number of manuscripts, the relations of which have not yet been thoroughly made out. the most recent critical text is that of c. hosius (2nd ed. 1906), and the latest complete commentaries are those of c. e. haskins (1887, with a valuable introduction by w. e. heitland) and c. m. francken (1896). there are separate editions of book i. by p. lejay (1894) and book vii. by j. p. postgate (1896). of earlier editions those of oudendorp (which contains the continuation of the _pharsalia_ to the death of caesar by thomas may, 1728), burmann (1740), bentley (1816, posthumous) and weber (1829) may be mentioned. there are english translations by c. marlowe (book i. only, 1600), sir f. gorges (1614), thomas may (1626), n. rowe (1718) and sir e. ridley (2nd ed. 1905), the two last being the best. (r. g.; j. p. p.)