Popular Science Monthly/Volume 36/March 1890/The Laws of Films
|←Absolute Political Ethics||Popular Science Monthly Volume 36 March 1890 (1890)
The Laws of Films
By Sophia Bledsoe Herrick
|The Psychology of Prejudice→|
By SOPHIE BLEDSOE HERRICK.
THERE is scarcely anything in the world which seems more utterly outside the realm of law than a soap-bubble. The delicate film, with its exquisite floating colors, its power of instantly vanishing, leaving no trace behind, hardly seems as though it could form a link in the inexorable chain of cause and effect which we call physical law.
The atmospheric pressure on a bubble six inches in diameter is over fifteen hundred pounds, and yet the fragile film lies safely between the opposing forces of nature — the pressure of the outer air, the spring of the inclosed cushion within it, the downward pull of gravity, the upward push of the buoyant atmosphere, and the molecular forces in the film itself: so long as the bubble lasts; it is because of an exquisite adjustment of all the forces, physical and molecular, concerned in its existence.
This is, of course, the merest commonplace, and yet it is one of the commonplaces of nature, which, however well we may know them, never cease to be wonderful when they are in any degree realized. There are other laws governing films which are no less wonderful, though they are less familiarly known. A heap of bubbles blown while the pipe is dipped under the surface of soapy water looks like a chaotic huddle of bubbles of all sizes and many shapes; but, upon careful examination, it is found that never more than three films meet at an unsupported liquid edge, and never more than four edges meet at a liquid point, and that the angles are always equal; that is, films will not meet each other at an unsupported edge or point at an angle smaller than 120° — one third of a circle.
Ordinary soap-suds made with clean hot water and ivory or THE LAWS OF FILMS. 621
pure Castile soap, and allowed to settle and clarify, or else filtered, answer very well for a series of simple and beautiful experiments in the forms assumed by soap films in order to fulfill this law of their union. There is a glyceric fluid made, which by various means has all the impurities of soap and water removed, and is toughened by the addition of pure glycerin ; and this is of course better, because much more persistent. A bubble made from this glyceric fluid, and carefully protected by a tumbler or bell-glass, will last for hours, and in some cases for days.
For these experiments in form, common suds last long enough to show the forms very satisfactorily, but for experiment in color the more lasting fluid is necessary, so a recipe for it is given here.*
When frames made in various forms, by bending fine copper wire, are dipped into the fluid, it is found that the films take on the most wonderful and beautiful shapes in order to fulfill the law of their union. The material of which the films are made does not at all affect their form.
With fine, well-straightened copper wire, outline a cube ; this may be done with the fingers or a pair of ordinary pliers, and the figure need not be very exact. The wires can be double along any of the lines ; let one end project from some corner for a handle, to be used in dipping the frame into the fluid.
Immerse this cube in the suds, and as you lift the frame out observe the films forming and shaping themselves. They usually take for a moment the form in Fig. 1 and slowly change to Fig. 2. Sometimes they retain the first form; in this case, the central drop with a glass of low power is seen to be not really a drop of fluid, but a tiny cube of films, each meeting the film from the wire edge by a curvature of its faces at exactly the required angle, 120. The films have constructed in their midst this tiny cube, because the twelve films could not otherwise meet in the center at the proper angle. This cube is formed and kept where a tiny bubble has been entrapped in the system of films. If no such bubble of air has been caught in with the films, they
- Plateau's mixture. This must be made in a warm room, temperature about 68 Fahr.
Let one half ounce of newly made Marseilles or pure Castile soap be dissolved in one pint (twenty ounces) of hot distilled water. When the solution has cooled to about the tem- perature of the room, it is filtered into a bottle. In using the common filter paper (bought at a drug-store for ten cents a dozen sheets), it is better to put only a small quantity of the mixture in at a time, and to support the paper in a funnel or muslin to prevent its break- ing. The paper soon clogs ; it should then be renewed. The process is slow, but not troublesome. When it has all been filtered, add fifteen ounces of pure glycerin either Price's or Scheering and Glatz's the ordinary glycerins are not fit for the purpose. Let the mixture be violently and frequently shaken ; then allow it to stand seven days ; on the eighth cool it to about 37 Fahr., and filter. If the liquid comes through turbid, pour it back and filter over again through very porous paper.
�� � assume the form of Fig. 2. The twelve films from the edges of the cube meet a square unsupported plate of film in the center.
With No. 2 still on the frame, dip it again into the suds. You catch a bubble by doing this which goes at once to the center (Fig. 3), and forms such a cube as existed at the center of Fig. 1,
|Fig. 1.||Fig. 2.||Fig. 3.|
only large enough to show the curvature of the films necessary to make them meet at their fixed angle. The laws of films formulated are as follows: 1. From each wire edge of a frame proceeds a film. Generally, if care be taken, no air will be inclosed, then every film will be in contact with the surrounding air on both its faces. 2. Only three films can meet at any liquid edge. 3. When several liquid edges terminate in one point in the interior of the system, the edges are always four in number, and the angles included between them are equal. 4. Whenever the films can fulfill these conditions, and remain plain films, they are so; when they can not, they are curved, but so curved that their mean curvature is null — that is, if in one part of the film the law of its union requires an upward curvature, in some other portion there will be an equal downward curvature to compensate for it.
In the films upon the cube frame, for instance, there is a slight curvature, just enough to enable them to meet each other on the angle of 120°. This is a very simple digression from the plane form, but in many other frames the divergence is very marked; for instance, in the triangular pyramid (Fig. 4) with wires dividing each side, after a bubble has been entrapped by a second dip, the curvature is very remarkable.
Plateau, the blind philosopher of Ghent, first studied this subject and formulated these laws. He began his studies with some experiments far removed from our films. In order to get some idea of the interaction of the molecular forces, he removed a mass of liquid matter he was observing, as far as he could, from the action of the physical forces. Using the well-known principle that a submerged body sinks till it has displaced its own weight of the fluid in which it is immersed, he made a mixture of alcohol and water of exactly the specific gravity of oil. Into the midst of this liquid he quietly introduced oil by means of a funnel. The oil lay passive between the equal downward pull of gravity and the upward lift of the alcohol and water. In this way the forces which bound the oil particles together had free play. The oil rounded itself at once into a sphere. For a time there was, of course, some chemical action between the oil sphere and the surrounding liquid; but, in making his observations, Plateau waited till these affairs had been settled between them, and their relations became fixed.
He then introduced into his oil sphere a rod, with a disk smaller in circumference than the oil sphere about it. Both of these were well oiled, and they entered the sphere without disturbing it. The globe of oil hung in the water, with the rod running through it in the position of the earth's axis, and the disk almost reaching to the line corresponding with our equator (Fig. 5).
By means of a handle the rod was turned, at first slowly, then gradually and steadily faster. The oil sphere slipped more easily around in its water socket than it would around the revolving rod and disk, and therefore turned upon its own axis. By varying his experiments, revolving his rod faster or slower, Plateau made a miniature representation of a world revolving about its own axis; he made his oil sphere throw off satellites, which revolved about the central sphere; he also, by what he calls a trick, imitated Saturn with its attendant ring.
He followed these experiments by using outlined frames of wire, such as we used for our soap films. These he adjusted around his hanging sphere of oil, and with a syringe withdrew the oil, making first a cube of oil with unsupported faces; and finally, as more and more oil was withdrawn, there resulted a system of oil films, each face of which was in contact with the water, exactly like those in Figs. 1 and 2.
This was the manner in which such systems of films were first reached; and, historically, the experiments have an interest in their relation to the subject of films as well as for the proof they offer that the material of which the films are formed has nothing to do with the forms they take on. Plateau went on from his oil films to those made with soap-suds and glyceric fluid. We have reversed the order in considering them, but it amounts to the same thing in the end.
Plateau's researches have been carried on by Brewster and others, and the subject much enriched by later experimenters. One of the most beautiful forms has not, it is believed, been published. A sphere is outlined with three equal circles, making, when joined together at equal angles, a globe with six meridians. When this is dipped in the suds, a rather complicated figure appears. It is sometimes necessary to dip this frame several times to get a perfect figure. From an axial edge of film three films start out. Just half-way between the axis and the outside curve of the sphere each of these three films meet two crescent-shaped films from two of the wire meridians, curved so that the three meet at the required angle. Sometimes when a bubble has been caught in the system, and always if a small bubble is carefully blown between two of the wires, a new figure will be formed. In an instant, as though the change were wrought by magic, the new figure flashes into existence. A long, six-sided, melon-shaped figure reaches from pole to pole inside the sphere; from each edge of this figure, entirely unsupported as it is by the wire, a crescent-shaped film reaches to each wire meridian.
|Fig. 6.||Fig. 7.|
The figures formed with the wire frames are usually perfectly symmetrical; but sometimes, from the peculiar form of the frame, symmetry is not consistent with a union at the angle of 120°. The law in such a case is obeyed, and symmetry cast to the winds. In Fig. 6, at the first dip the figure is very unsymmetrical, though always the same. When a bubble is blown on the bottom, the figure starts out perfectly symmetrical in form.
Brewster has added many experiments to those of Plateau's. The next one given is his, and a very curious one it is too. Two rectangles are made of the copper wire; one is slipped within the other and held at right angles to it; they are in this position dipped into the suds. The system which starts into being can be seen in Fig. 7. The central oval stands diagonally just half-way between two of the angles made by the crossed frames. Now, if the frames are gradually turned upon each other, which it is very THE LAWS OF FILMS. 625
easy to do, the form of the oval changes. At right angles the oval film, is four times longer than it is broad. As the angle between which the oval film stands is increased, it widens till it is nearly square. If the rectangles could be made to lie exactly one upon the other, the oval film would fill up the space. Now, when the angle of the two wire frames is made narrower instead of wider, the oval narrows till, at 45, it is a line, and in one moment the system has changed : the oval stands between the wider angle just across its old position and at right angles to it.
A still more remarkable change takes place when a bubble is blown upon the oval film, the lines being at right angles to each other. When it reaches the proper size, all the films disappear, and a hollow curvilinear cube is formed, each side curving out from the wires which define its vertical edges. At the top and bottom the wires make a cross on the film ; in each of these triangular spaces four summits appear ; colored rings form around them ; a black spot shows in the center of each summit, and the bubble bursts. If the wires are held straight up and down when the bubble bursts, the old system of films will start into being again, as if it had left its ghost behind it to recover the elements which the bubble had appropriated.
Dr. Sloane, in his " Home Experiments in Science," gives some beautiful figures. A wire is bent in a spiral, with one end turned straight up through the middle like an axis. Dipped in the fluid, it gives a single spiral film curving around the central wire as a spiral staircase curves around its central pillar. He also gives some very simple and interesting experiments showing the traction of films, requiring no special apparatus or fluid, and so within the reach of every one. All the frames used in this article were made of thin copper wire bent into shape with the fingers or a pair of pliers. Of course, if the wires are soldered instead of being twisted together, and are covered with a thin film of par- affin by rubbing a so-called wax candle on them and then holding the frame above but not too near a bed of coals, the films will last longer ; but that is the only difference.
The wonderful traction of films is shown by the recent experi- ments with oil upon the waves in a storm. The oil, of course, does not still the waves, but it converts the combing waves, so dangerous to navigators, into a comparatively harmless swell. It is the traction of the film which prevents the wind from drawing the water up the incline of the wave and sending it jetting up- ward to fall over in a comb. A film of oil ^-oisVo-o of an inch in thickness will hold the wave of water driving before a gale so that it can not break into spray.
The closing words in Brewster's experiment on the revolving rectangles of wire bring us to another remarkable though famil-
TOL. XXXVI. 40
�� � 626 THE POPULAR SCIENCE MONTHLY.
iar fact in regard to films their wonderful and changing colors, what in scientific language is called " the colors of thin plates/' because the same effect is produced in many cases where " film " is not exactly the word to use.
The colors of soap-bubbles furnish one of the most triumph- ant vindications of the wave theory of light, and offer to its opposers one of the hardest possible nuts to crack. A brief expla- nation of the wave theory of light and interference, so far as it bears upon our subject, will, it is hoped, be pardoned. It is an idea so familiar to those who have studied physics, and yet so difficult of conception to those who have not, that a few words seem neces- sary in a popular exposition of the colors of films.
Light is, of course, our name for the sensation, but back of the sensation there lie the physical conditions which are its cause. The theory of Newton, that light is caused by minute particles of matter shot out from the luminous body, stood the test of the simpler phenomena ; but, when it came to the explanation of soap- bubble colors, his theory, even with the marvelous ingenuity which he brought to bear upon it, broke down. If light were matter, it is impossible to see how one light can be added to another light and produce darkness, which is sometimes the case-, while, if it were motion, we can readily see how motion may be added to motion, and the result be rest. A sound can be so added to a sound as to produce silence, but the more familiar illustration is with waves of water. Two stones dropped into water will pro- duce waves, and where these meet there are points at which the water remains at its original level. This is because at these points one set of waves tends to raise the water while the other set tends to lower it, and between the two it remains where it origi- nally was. This occurs in some parts of the ocean where the tidal wave sweeps around an island and meets, one wave being half a length behind the other, in which case they simply neutralize each other. Where the crest of one wave would have been, the trough of the other would have been at the same time, and be- tween the two impulses in opposite direction at the same moment the water remains unmoved, and there are no tides.
Darkness corresponds with this unmoved plane of water, and with silence in the case of sound-waves. If light were simple waves, as a result of such interference we would simply have darkness, and as a result of partial interference we would have all the gradations from darkness to light ; but a light-wave is not a simple undulation, it is made up of innumerable vibrations of various wave-lengths, each of which corresponds with a color or tone. The resultant of all these motions combined is white light. Extinguish one rate of vibration, say the smaller waves which cause the sensation of blue, and we have a wave the resultant of all that
�� � THE LAWS OF FILMS. 627
is left behind, which will be yellow. Color is a partial extinction of light not of light as a whole, but a suppression of one of its constituents. If you take a yellow glass and allow the light to fall through it, you will find it transparent ; in the same way a blue glass is transparent ; but if these glasses are the comple- mentary blue and yellow color, and placed one on top of the other, no light comes through them. The yellow glass sifts out all the blue rays, and the blue glass sifts out all the other rays, and no light can get through. If the colors are not pure, it is usually because the yellow has some green in it, and so has the blue. Neither the yellow glass nor the blue is competent to sift out these rays, so we see green come through them both. This is the case in mixing blue and yellow in paints: the resulting green does not come from the mixture, but is the sediment you might almost call it left after the pure blue and pure yellow have neu- tralized each other.
It is clear that, if two waves can be made to set into vibration the same medium at the same time, and from almost exactly the same center, one of them being a half-wave or several half- waves' length behind the other, we shall have, as in the case of water and sound, no movement, or darkness. If there is not exactly a half-wave's distance between them, some color-waves will neutralize each other and be extinguished, and we shall get the complementary color the resultant of all that is left un- neutralized.
This is the cause of all the flitting and changing colors in soap- bubbles, mother-of-pearl, peacocks' plumage, opals, and iridescent glass. By some means certain vibrations have been extinguished by interference, and we see the resultant of the rest. Whenever light goes from one medium into another, even when both media seem perfectly transparent, there is a partial reflection from the surface where the media meet. Hold a pin against the surface of a piece of glass (unsilvered plate glass is the best) : you will see two faint reflections of the pin, one from the front surface of the glass and one from the back, and yet the main part of the light reflected from the pin goes through, as you can easily tell by looking through the glass at the pin. So it is with a soap film : when light falls on it, most of it goes through, but there is a slight reflection from the outer surface of the thin lamina of soap-suds and another slight reflection from the back of it. The two sets of reflected waves start from points so very near each other that they both act on the medium in different directions at the same time and in the same place, and we have color.
If light went forward like a regiment of soldiers in line, there might be just as much interference from the plate of glass as there is from the film of soap-suds ; but it does not it goes out in
�� � 628 THE POPULAR SCIENCE MONTHLY.
circular or, rather, spherical waves in every direction from the starting-point. It is only waves of light which are reflected back from two points very, very near each other, which produce the colors of interference. Circles which do not have the same cen- ter cut each other only at two points ; but, the nearer the two centers are, the more nearly the circumferences coincide. When the light comes back colored from a piece of mother-of-pearl, it is because the waves are reflected back from lines so close together that you can not see them, except under a very high power of the microscope, and so they interfere. Metal may be ruled with lines that give back the same sort of color, and perfect impres- sions in black sealing-wax of the colored pearl will show colors in the same way.
The colors which flit over the surface of a soap-bubble each tells the story of the thickness of the film at that point. These films are exposed to the movement and drying effects of the air, and to the irregular puffs of air entering from the mouth in blowing them ; but if a film can be secured from these influences and allowed to become gradually and evenly thinner, even and regular colors appear. Blow a soap-bubble in a watch-glass filled with the soapy fluid. Let it sit in a saucer in which there is also some of the fluid, and cover with a clear glass tumbler the instant the bubble a little overhangs the watch-glass. The soapy fluid in the saucer prevents the air from getting in or out of the tumbler. Such a bubble blown from soap-suds made of distilled water and white Castile soap, which had been standing a very long while and become crystal clear, lasted for three hours and a quarter. It had no colors upon it when covered. They began to form at once : broad bands of pink and green slipped down from the apex ; then came closer and more vivid rings of color ; at last a black spot ap- peared, which grew in size. In the long-lived bubble just spoken of, the whole upper part became a metallic gray, covered with clouds of darkness and velvety black spots, the colors being crowded from the apex down to the edges. That these appear- ances are all due to interference is proved by the fact that, when the light by special means is prevented from reflection at one of the surfaces of the film, the color disappears.
There is no special advantage for home experiments in having a bubble last so long. Very much the same changes occur in a bubble which lasts for half an hour as in one that lasts for three hours, only they occur more quickly.
The colors of films are rarely, if ever, pure prismatic colors ; they are the resultant of certain colors left after the extinction of others. Various shades of green, from almost gold to the intensest emerald green, orange dusky with red, red magenta-colored from the admixture of blue waves, and so on, are the colors seen.
�� � Another simple and very interesting experiment is to twist a copper wire into the shape of a tennis-racket or battledore, dip it into the fluid and set it upright under the tumbler. If the saucer is partially filled with yellow beeswax, melted and allowed to harden, this can be very easily done. The colors in this case come down in bars, in the same order as they did on the bubble; the black spot is much larger and more irregular in shape. In one instance, with a simple soap solution, this spot of intense black covered three quarters of the frame before the breaking of the film. Many films may break before one is secured which will last so as to show these effects.
The cause of these regular rings and bars of color is that the film gradually thins from the top, by the slow streaming off or evaporation of the suds from the film, and for each definite thickness a definite color appears. The black spot which comes last of all shows that the film at that place is just one half a wave-length of light in thickness, a size entirely too small for our conception, though it can be told in numbers. The length of a wave of red light is about 1/37000 of an inch, and of all the other colors smaller.
The circulation and changes in the film are most curiously revealed by the movement of flecks of color on its surface.
There are other ways of making inequalities in the film, which are revealed by the colors. A little instrument, called the phoneidoscope,
Fig. 8. — Phoneidoscope.
A, bell-glass; B, elbow; C, India-rubber tube; D, wire support; F, upper half of mouth-piece; E, lower half of mouth-piece; G, diaphragm.
which may be either bought or very easily made, shows most beautiful figures which start into shape in answer to musical notes sung or words spoken into it. It is in all its forms a modification of, or improvement upon, this idea: an inch tube of India rubber of any length, with a funnel on one end and a mouth-piece on the other, diaphragms of thin metal or varnished cardboard being placed across the mouth of the funnel with holes of various shapes cut in them to sustain the film. A very satisfactory one may be made with very little trouble and at slight cost: three feet of inch rubber tubing, a bell-glass, such as is used to shade night tapers, some pieces of cardboard or thin brass, which can be cut with the scissors, and an inch tin elbow, used in speaking-tubes and costing three cents. Fit the parts together as in Fig. 8. The diaphragms should be blackened and varnished if of cardboard; the holes in them can be triangular, square, round, or of any geometrical shape. A film is drawn across the hole in the diaphragm; it should be set upon edge till the colors are established, then it is to be laid across the mouth of the bell-glass, and into the other end of the tube notes can be sung; but the breath must not be inhaled or exhaled carelessly, or the film will be broken.
A closed mouth-piece may be made by filing off two tin toy trumpets two inches from the open end. Over one tie a stretched membrane of India-rubber sheeting, such as dentists use, or fasten
Fig. 9. — Figures on Films in Phoneidoscope.
A, B, C, forms whirling and evanescent; D, E, F, forms which remained for some time after the vibrations of film ceased.
with paper a thin sheet of mica, E, or even tough, strong letter-paper may be used. Hold the second trumpet, F, reversed against this, and sing into it.
The colors and figures on these films, if one is patient and learns how to use the voice, are simply incredible they are so wonderful and gorgeous. Fig. 9, A, B, C, D, E, F, show the forms obtained on several diaphragms with the home-made phoneidoscope described above, some with the closed mouth-piece and others by simply carefully singing into the rubber tube. The colors change constantly, and are so rich and gorgeous that they seem to have lost their transparency and to be metallic plates, except for their streaming, swirling colors. The diaphragms should, as was said before, be blackened, and the film seen projected against a black background. I simply use a piece of black material placed behind and a little below the bell-glass on which the diaphragm rests.
These interference colors do not require a film of any special substance, or, indeed, of any substance at all. The air between two plates of clean, clear glass, pressed together and worked with the fingers till they are as close as possible, gives beautiful rings and fringes of color. A crack in the center of a block of clear ice, where there is not even air, but only empty space, or, rather, the ether that fills all space, gives out gorgeous colors of interference.
The colors of iridescent glass are due to interference. In its manufacture, by some chemical means, the surface film has been made different from the glass below, and so acts as the soap film does, and gives out its lovely tints. A drop of turpentine on the surface of water on a black tray shows fringes of color from the same cause.
One of the most beautiful examples of interference color may be seen at the Metropolitan Museum of Art in New York, in the Cesnola collection of ancient glass. Originally this glass was evidently ordinary transparent glass. From some cause the surface has been acted upon till it lies in thin films one upon the other, sending back to the eye the most gorgeous interference colors. By the courtesy of Dr. Isaac Hall, the curator of the museum, I was enabled to examine some fragments of this glass microscopically.
The whole surface is made up of a series of films of the most exquisite delicacy. There are tiny cavities united by a network of lines from which the decomposition has spread laterally in every direction. Flakes come off with the lightest touch, so thin that it seems impossible they should be capable of subdivision, and yet a good two-thirds glass (about one hundred diameters) shows it to be made up of a number of superposed plates. The fact that the color of this glass is due to interference is proved by putting a drop of alcohol or oil upon a flake, when the colors disappear or are entirely changed. As the liquid dries, the colors gradually come back.
The beauty of this glass under the microscope is simply indescribable. Gold and silver, exquisitely wrought, and vivid with every known jewel, would be tame and colorless beside it (Fig. 10, A). The films, as they come off, are in many cases not ordinary flat films; the outlines are sometimes very singular, being made up of most eccentric curves in all sorts of combination. In one or two instances these scales overlapped, showing that the disintegration had taken place in a spiral direction (Fig. 10, B). This world of beauty, in both colors and form, was found within the area of one square inch or less, on a small fragment of no special brilliancy to the naked eye.
Brewster describes, in the "Transactions of the Edinburgh Royal Society," some specimens A of ancient decomposed glass, but they must have been in a much earlier stage of decomposition than the Cesnola glass, judging from the figures and descriptions given. He states
Fig. 10. — Cesnola Glass.
A. a, emerald-green, with strings of bubbles light-green and brilliant, like pale emeralds; b, bronze-gold ground, spots of violet, and bronze-gold rings, ruby, pale vivid blue, and deep sapphire blue; c, partly scaled film, vivid violet, toning down, with spots as above; d, deep violet-blue, like the sky on certain nights; e, speckled gold; f, exquisite violet, with bubbles like pearls, only shaded violet tone.
B. Shape of violet layer as it came off, very thin.
that the experiment had been made of submitting glass to powerful solvents, when, in a short time, circles and other forms, centers of decomposition, began to appear. Here was probably the suggestion which has since been followed in the manufacture of our modern iridescent glass. In a piece of iridescent glass, brilliant at first, but which has been growing more brilliant for several years, I find a number of distinct centers of disintegration, showing the process, whether by art or by time, to be identical in kind.
The question involved in the problem of air navigation is regarded by Mr. E. N. Lewis as simply one of increasing power without increasing the weight of the apparatus by which the power is applied. The supposition that the vehicle must be lighter than the air, on which experiment has mostly proceeded, is a mistaken one. "A bird can fly, not because it is comparatively light in weight (for it is not), but because it is strong." The successful air-ship will be a large structure, very light in weight compared with its strength, but many times heavier than the air it displaces, and propelled by machinery capable of developing enormous power. "The skill which has produced . . . the modern bicycle will not find the task of designing such a structure too difficult."