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The method of limits of D'Alembert, which is now more fro quently used than any other, was considered by the author himself as an explanation of Newton's prime and ultimate ratios. It is usual to attribute the exposition of this method to D'Alembert (and, considered as an actual application of limits, correctly), though several, previously to him, had made special applications of the principle. The articles in the Encyclopédie must be considered as the pièces justificatives. The following article, DIFFERENTIAL COEFFICIENT, will explain this method, which contains the point in which the principles of all the preceding unite, and which must more or less be in all. See also the article LIMIT.

The two remaining methods (those of Landen and Lagrange) are attempts to establish the science upon purely algebraical principles. Previously to entering upon them, we must remark that none of the preceding theorists attempted to make his system furnish any additional security to the methods of the algebra already in use. Such as it was, correct or incorrect, clear or obscure, no one gave a moment's consideration to the fact that algebra already contained difficulties of precisely the same character as those which were matter of dispute in the differential calculus. Taking it for granted that algebra in every part stood already upon as firm a basis as the differential calculus could ever on any supposition be expected to do, it was a matter of some interest to make the latter a pure extension of the former.

Morgan's treatise on the subject, published by the Society for the Diffusion of Useful Knowledge, professes to have the same end in view. The works of M. Cauchy have much in them by which this object is promoted, but expansion is avowedly introduced.

We shall now state two propositions, one geometrical, the other algebraical, in the words of the several systems.

1. Infinitesimals.—An infinitely small arc of a circle is equal to its chord.

Prime and Ultimate Ratios.-If an arc of a circle diminish, the ultimate ratio which it bears to its chord is one of equality; or if it begin to increase from nothing, the prime or nascent ratio of the arc and chord is that of equality. Or the arc is ultimately equal to its chord. Fluxions. If an arc increase from nothing with a uniform velocity, the velocity with which the chord increases is, at the first moment, equal to that of the arc.

Limits.-If the arc of a circle (and therefore its chord) diminish without limit, the limit of the ratio of the arc to the chord is one of equality. arc 0 = 1, chord 0

=

Residual Analysis.-When the arc of a circle 0 which is ascertained by clearing the numerator and denominator of

= 0.

a factor which vanishes when arc =
Theory of Functions. When the arc is expanded in the following
series,
Arc A x chord + B x (chord)2 + &c.
then A = 1.

2. Infinitesimals.-If an infinitely small increment de be given to x,
then 3 receives the infinitely small increment 3x2dx.
Prime and ultimate Ratios.-The ratio which any increment given
tox bears to the consequent increment of 3 is ultimately that of
1 to 3 x2.

Fluxions.—If x be a line which increases with a velocity, then 23 increases with the velocity 3xx.

The residual analysis of Landen is a technically algebraical exhibition of the theory of prime and ultimate ratios. The tract in which it was promulgated, 'the Residual Analysis, a new branch of the Algebraic Art,' &c., appeared in 1764. When it is considered that this new branch of the algebraic art was only old fluxions in a different dress, the title may excite surprise, if we remember how well Landen deserved his reputation. But it must be remembered that all the discussion of which this article is meant to elucidate the history, arose from a tendency to consider two methods as mathematically different, which were not the same in the method of enunciating their first principles. A something between Landen and D'Alembert, as to principle, published in 1748, was called the Doctrine of Ultimators, containing a new Acquisition, &c., or a Discovery of the true and genuine Foundation of what has hitherto mistakenly prevailed under the improper names of Fluxions and the Differential Calculus.' The difference between Landen and Newton will appear in the article FRACTIONS, VANISHING, and in the instances which we shall presently give. It is the LIMIT of D'Alembert supposed to be attained, instead it follows that when y=x, of being a terminus which can be attained as nearly as we please. A little difference of algebraical suppositions makes a fallacious difference of form; and though the residual analysis draws less upon the dish, putable part of algebra than the method of Lagrange, the sole reason of this is that the former does not go so far into the subject as the latter.

The method of Lagrange, first given in the public lectures at the École Normale, and afterwards published separately under the title of Théorie des Fonctions, is a deduction of the whole science from Taylor's Theorem, which being absolutely granted, undoubtedly all the rest may be made to follow. If p (x+h) can be always expanded in a form of which the two first terms are x+o'x. h, and if '(x+h) be related in the same manner to p'x+o'x. h, and p''(x+h) to p'x+p'"x. h, and so on, it can be made to follow that h2 2

13

(x+h) = x + p'x. h + p′′x.・+"'x· 2.3" + &c. upon principles as sound as those of algebra in the hands of Maclaurin, or Euler, or Clairaut, as elementary writers. It is our opinion that Lagrange has not been correctly understood, nor fairly dealt with, by those who have compared his theory of functions with the other methods. Undoubtedly any one who should maintain the unqualified admissibility of Lagrange's work must assert both the major and minor of the following syllogism.

Algebraical expansion (théories des suites is Lagrange's phrase) as generally received in 1790, was founded on sound principles: the théorie des fonctions is a logical and incontestable result of such algebra; therefore, &c.

All the attacks upon Lagrange have denied the major of this syllogism, whereas it appears to us that he never intended to assert more than the minor. Perceiving that the mathematical world was in the habit of calling in the aid of limits or infinitesimals, to help a certain algebra in deducing certain conclusions, he showed them how that very algebra, good or bad, was competent to the deduction of the same conclusions without either limits or infinitesimals; and he was correct. Notwithstanding anything in the work in question, Lagrange might have admitted all that we shall find it necessary to say against his system, absolutely considered, in the article FUNCTIONS, THEORY OF.

A new question has arisen of late years, namely, whether the theory of limits be not absolutely necessary to the rigorous development of common algebraical forms, and whether this same theory of limits may not be applied to the establishment of the differential calculus, independently of any expansion. A tract of M. Ampère (as we believe), entitled 'Précis du Calcul Differentiel,' &c., is the earliest writing we are acquainted with in which this is attempted to be done. Mr. de

Limits. The limit of the ratio obtained by dividing an increment of 3 by the increment of which produced it, on the supposition that the latter increment diminishes without limit, is 3 xa. Residual Analysis-Since

y3-23
y-x

= x2 + xy + y2
3x2.

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Theory of Functions.-If (x+h)3 be expanded in a series of powers of the coefficient of the first power of h is 3x2. DIFFERENTIAL COEFFICIENT. The expressions to which this term is applied are of a degree of importance in the science to which they belong, as great as that of the letters of the alphabet in writing. Without entering into the method of using them, which would be in effect to write a treatise on the differential calculus, we shall make some remarks on the manner of defining and understanding the term.

When two magnitudes are so related that either being given the other is also given, it follows that any change being made in the one, the consequent change in the value of the other can be found, and the two changes can be compared as to magnitude. By this means a rough notion can be formed as to the effect which a change of value in one produces in the other. In such articles as CURVATURE, DIRECTION, VELOCITY, FORCE, &c., it is sufficiently shown that this rough notion, obtained by making a sensible change in one magnitude and comparing it with that produced in another, though sufficient for practical purposes, does not afford any exact and mathematical measure of the thing sought for. In each of the articles cited, it is found necessary to diminish the change originally supposed without limit, and it is not the actual ratio of two changes which we have to consider, but the limit to which that ratio ap roximates as the changes are diminished without limit. We must distinctly refer the student to sensible objects for an illustration of the cause why it is convenient, and even necessary, to have recourse to the limit of a ratio. [See also DIFFERENTIAL CALCULUS.] LIMIT, RATIOS (PRIME AND ULTIMATE), and various articles cited in

Let or be a function of x, called y, which when x is changed into x+h, becomes y+k, so that

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k = (x + h)2 + (x + h) — (x2 + x)
= 2 xh + h2 + h;

and k divided by h gives 2x + h +1, which, when his diminished
without limit, has for the limit of its decrease, 2x + 1, which is there-
fore the differential coefficient of +x with respect to x.

The term differential coefficient arises thus. The method of Leibnitz [INFINITESIMAL CALCULUS] proceeds as follows: Imagine in the preceding expression to be increased by an infinitely small quantity (see

article just cited for remarks on this phrase) which call dx, the diffe- outline with the body, except at its angular points; the order of the rential of x. Then the resulting differential of y is

dy = (x + dx)2 + (x + dx) — (x2 + x) (2x + 1) dx + (dx)2

=

Now (dx) is rejected as being an infinitely small part of the preceding term, so that (2x + 1) de is the differential of x2 + x. expression 2x + 1 (which is our preceding result) is here the coefficient The of the differential dr, and when a less objectionable method of obtaining it came into use, still retained the name of differential coefficient. were exceedingly to be wished that some shorter term could be agreed on for the expression of a result which is so often required to be named.

It

In the method of Leibnitz the differential coefficient is actually dy dy divided by dx, and it is still expressed by in the more modern dx methods. But this notation must now be supposed to be obtained as follows: Let the change of x into x + Ax be accompanied by that of y Ay into y+Ay; then expresses the algebraical ratio of the change of dy dx

Ax

y to the change of x. In making the convention that Ay Ax'

is to stand

for the limit of obtained from the supposition that Ax diminishes without limit, we consider the first symbol, not as that of an algebraical fraction, but as one whose whole has a meaning, the parts having none. This is the case with all simple symbols, as distinguished from compound symbols: thus in the figure V, standing for five, the two sides of the letter have no independent meaning; while in 4 +7, each of the three symbols, 4, +, and 7, has meaning contributing to that of the whole, and at the same time independent of it. dy Thus though we consider as pointing out that it is y which has been differentiated, and with respect to x, we do not, in this symbol, assign to dy and do any independent share of meaning.

dx

The advantage of the notation in question is the connection which it preserves between the practical use of the method of Leibnitz, and the theoretical accuracy of that of limits. For instance, it is strictly true, according to our conventions, that

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but, giving dy and de separate meaning, it is not true that dy = (2x + 1) dx. Yet the latter equation, though never true, is of this kind, that it may be made as near to truth as we please, if dr may be made as small as we please. For the reason why it cannot therefore be said to be true in the case where dx = 0, see FRACTIONS, VANISHING.

The differential coefficient of a differential coefficient is called the second differential coefficient; a repetition of the process gives the third differential coefficient and so on. These, if the original method of notation were proceeded with, should be represented by

:

dy

dx

dx dx

313

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dx

dx

colours in each fringe, reckoning from the outside towards the shadow, or inside, is, as in the prismatic spectrum, from red to blue, but the intermediate colours are less distinctly isolated, partaking of a mixture of the extreme tints.

The actual shadow, or dark space within the innermost fringe, is if the rays had passed exactly by the edge of the body in straight lines also larger than the geometrical shadow which would have been cast and been received on the same screen, and the illumination instead of shadow, extending indeed to a small amount even within the geobeing cut off sharply, fades continuously into the darkness of the metrical shadow. The space by which the actual shadow surpasses of the interposed body, and likewise, except in the case of a very the geometrical in breadth, is independent of the form or the matter slender body, of its linear dimensions. The dilatation of the shadow will therefore be most striking in the case of narrow bodies such as pins, hairs, &c.

When the interposed body is very narrow but of sensible width, streaks alternately brighter and darker will be found within the shadow, and a white line along the middle, when the body is of a long and slender rectangular form.

All these phenomena are seen more easily and with greater brilliancy if the light, instead of being received on a screen, be admitted directly into the eye armed with a lens or eye-piece.

If the incident light were homogeneous, such as pure red, blue, &c., as found in the spectrum, the colour of the fringes would of course be the same as that of the incident light, and the fringes would consist of simple alternations of light and shade. It is found that the scale of the fringes varies with the colour, the fringes being broadest for the least refrangible colours. fringes are variously coloured, in consequence of the encroachment of Hence when the incident light is white the the bright and dark fringes belonging to the various simple coloured lights of which, as we know, white light is compounded.

Sir I. Newton in his optics relates several modes in which he varied the experiments on the inflexion of light; and, in his queries, he suggests an explanation that light may be subject to the action of forces sensible only at very small distances from the surfaces of bodies; that under their influence the rays in their course near the edge of the body, instead of passing straightforward, describe very sinuous paths, with many contrary flexures, and thus they are found some turned inwards towards the shadow, others outwards, forming coloured bands; the curve surface, which is the locus of the intersections of all those diffracted rays, forming the envelope of the visible shadow. Little advance has been made beyond Newton's conjectures towards explaining, according to this doctrine, the curious phenomena of diffraction; and the explanation, incomplete as it is, has been shown by Fresnel to be subject to serious difficulties.

Very different has been the result with the theory which makes light to consist of undulations. This theory was first applied to the explanation of the phenomena by Dr. Young, who employed the principle of interference, the discovery of which we owe to him, and which he so successfully applied to the explanation of the colours of thin plates. The bending of light around an opaque body was assimilated to the bending of waves around an obstacle, and the internal fringes in the shadow of a narrow body to the interference of the light so bent round the opposite edges. That these fringes are incontestably produced by interference, he proved by cutting off the light passing either

but a more convenient notation is derived as follows: Let the sub-edge by means of an opaque screen placed either before or behind the stitution of x + Ax for x take place any number of times in succession, giving

• (x), ¢ (x + ▲x), ¢ (x + 2 ▲ x), 4 (x + 3 ▲ x), &c. Form the successive differences of the first term [DIFFERENCE] px or y: then the following theorem can be proved. The nth difference of y, or Ay, divided by the nth power of Ax, or (A)", is an expression of which the limit, made by diminishing Ax without limit, gives the same result as arises from differentiating y, n times in succession. We may therefore (subject to the remarks already made on the first coefficient) represent the limit of (Ax)" so that the function y and its successive differential coefficients are denoted as follows:

dry

by (dx)"

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DIFFRACTION OF LIGHT. The peculiar modifications which light undergoes when it passes by the edge of an opaque body are classed as phenomena of the diffraction or inflexion of light.

When a pencil of solar light is admitted into a darkened room through a very small hole in a card, or is collected in a point by means of a convex lens of short focus, and then diverges from that point, if a small opaque plate of any outline be interposed in the course of the ray, the shadow of this object received on a parallel screen behind will be encompassed by a series of coloured bands or fringes of a similar

external fringes were attributed to the interference of the direct light narrow body, when the internal fringes immediately disappeared. The with the light reflected from the edge of the diffracting body (Phil. Trans.' for 1802, p. 12). Several of the observed laws of the phenomena were shown to be explicable on this hypothesis.

This explanation was in the first instance adopted by Fresnel; but in pursuing his researches he met with phenomena which opened his eyes to the insufficiency of the explanation, and at last led him to perceive that the various observed appearances flowed naturally from two grand principles-the principle of interference, and the principle of Huygens-which themselves are but particular applications of the general dynamical principle of the co-existence of small motions. In his celebrated memoir on diffraction (Mém. de l'Académie,' 1821, tom. v. p. 339), he has followed out this theory into its mathematical details, and compared the results with observations both qualitatively and quantitatively, by a series of most careful measures; and the agree ment is complete. The theory furnishes not a single disposable constant whereby a seeming accordance between theory and experiment might have been brought about, for the length of a wave of light, the only unknown constant involved, admits of being determined independently of diffraction by the interference of two streams of regularly reflected or refracted light.

About the same time, and independently, Fraunhofer was engaged in a series of researches on diffraction of a somewhat different class, namely, those produced when a luminous point or slit is viewed in focus through a telescope, and the object-glass is covered by a screen pierced with one or more apertures. The reader may easily observe phenomena of this kind by viewing a bright point, such as the image of the sun in the bulb of a thermometer, and covering the eye with a piece of fine gauze, or of tin foil or black paper pierced with one, two,

627

DIFFUSION.

or three minute needle-holes, or apertures of other shape. The most remarkable instance observed by Fraunhofer was that of a fine grating through which was viewed a slit of light parallel to the lines of the grating. In this case spectra were formed so pure as to exhibit the fixed lines; and in this way Fraunhofer has measured with the most extraordinary accuracy the wave lengths corresponding to the fixed lines which he denotes by C, D, E, F, G, H. (Gilbert's 'Annalen.' Bd. 74, s. 337.) This class of phenomena has been most elaborately investigated by M. Schwerd (Beugungserscheinungen, &c.).

DIFFUSION. This term has a somewhat extended application in modern science: thus we speak of the diffusion of light, and of heat; the diffusion of gases and of liquids: we also speak of a diffusive power. The diffusion of light is intimately related to its reflection. If we accept the optical definition of reflection, according to the law of equal angles, it would be impossible to account for the visibility of objects: the geometrical reflection would perfectly represent to the eye the source of the incident beam, but it would convey no idea of the size or colour of the reflecting surface. According to this law the image of the sun would be reflected from the surface of bodies just as if they were perfect mirrors, but the bodies themselves would not be seen the moon would appear like a mirror reflecting the image of the sun: the candles in a room would be reflected by the walls and other objects, and nothing would be seen but the multiplied reflections of the flame. But as the surfaces of bodies are not perfectly reflective, but on the contrary, contain innumerable roughnesses which absorb a portion of the light and scatter the remainder, each roughness becomes the focus of a pencil of reflected light, the rays of which diverge equally in all directions, bearing with them the colour of the reflecting body. Thus each point of the surface forms an independent focus, by means of which each such point is made visible. This is called, somewhat improperly, irregular reflection, but there is nothing irregular in the phenomenon, for by means of it, light is diffused, and the illuminated objects become as it were secondary luminaries, and render objects visible which are not within the direct influence of the source of light, and these objects in their turn, by reflecting light irregularly illuminate others, and in this way the light is reverberated backwards and forwards so as to render objects visible which are far removed from any direct source of light. By means of this same property the atmosphere diffuses the light of the sun in all directions, and produces the phenomena of twilight. Similar phenomena are also to be noted with respect to heat, although it is not easy to determine the effects with as much precision. Radiant heat obeys the law of equal angles, but a portion of it that is not absorbed must be diffused by irregular reflection. This subject will come again under our notice in the article HEAT, and we content ourselves here with this slight indication of it.

The term diffusion as applied to gases and liquids, refers to that process by which such bodies when in contact, pass through each other and intermingle, although not necessarily related by chemical affinity. It was noticed by Priestley, that if a closed bladder containing oxygen gas be placed in a jar of hydrogen gas, the jar and the bladder will after some time each contain explosive mixtures of oxygen and hydrogen. This phenomenon, however, is now referred to endosmosis, or the osmotic force. [OSMOSE.] As an example of gaseous diffusion, we may instance the case of chlorine and hydrogen, the one being thirtysix times as heavy as the other, yet if a bottle of hydrogen be inverted, and connected by means of a long tube with a bottle of chlorine, the heavier gas will in a few hours creep up the tube to mingle with the hydrogen, and the light hydrogen will descend the tube to mingle with the chlorine, and this mixture will be equal in every part of the apparatus. Indeed, if sufficient time be allowed, this equal intermixture will take place between all gases and vapours which do not act chemically on each other. The rapidity with which this diffusion occurs varies with the density of the gases, and the more widely they differ in this respect, the more rapid is the diffusion. The rapidity with which a light gas diffuses into a heavy one may be shown by means of a diffusion tube, consisting of a glass tube 12 inches long, one end of which is closed with a dry porous substance, such as a plug of plaster of Paris: when this is filled with hydrogen gas at the pneumatic trough, and supported so that the water shall stand at the same level on the inside and on the outside, water will be seen to rise up the tube, and will continue to do so in opposition to gravity, until in a few minutes, it will be some inches higher than the water outside, in consequence of the gas escaping through the pores of the plaster, and diffusing itself into the air more rapidly than the air passes in and diffuses itself through the hydrogen. Experiments made in this way show that the diffusiveness of a gas is in the inverse proportion of the square root of its density. The density of air being 1, 1, and its diffusiveness = 1, the the square root of that density density of hydrogen 0-0692, the square root of that density 3-7994, or in an actual experiment 0.2632, and its diffusiveness 0-232 3-83. That is to say, while one measure of air is passing into the diffusion tube, 3.83 measures of hydrogen gas are passing out of it. When different gases are introduced into the diffusion tube, each maintains its own rate of diffusion. The rate of diffusion is increased by a rise of temperature, but not so rapidly as the direct expansion by

heat.

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The process of diffusion performs an important part in nature.

DIFFUSION.

"Accumulations of gases, which are unfit for the support of animal
and vegetable life, are by its means silently and speedily dispersed,
and this process thereby contributes largely to maintain that uni-
formity in the composition of the aerial ocean which is so essential to the
comfort and health of the animal creation. Respiration itself, but for
the process of diffusion, would fail of its appointed end, in rapidly
renewing to the lungs a fresh supply of air in place of that which has
which it has undergone." (Miller, Elements of Chemistry,' part i.)
been rendered unfit for the support of life by the chemical changes
The escape of a gas through a minute aperture into a vacuum is
called EFFUSION, which see; and the passage of gases through capillary
tubes into a rarified atmosphere belongs to TRANSPIRATION, while the
passage of gases through diaphragms must be referred to endosmosis.
[OSMOSE.]

Liquids diffuse themselves among each other, except in those
In most cases
few cases where they are not miscible, as in the case of oil and
water, or but little miscible, as ether and water.
however, the adhesion between the particles of dissimilar fluids is
very perfect, so that they become completely incorporated; so much
so, that an actual penetration of one body by the other seems to
take place, since the mixture occupies less bulk than the liquids did
separately. Thus 100 parts of alcohol mixed with 100 of water
will measure only 196 parts, and the same proportions of sulphuric
acid and water will measure only 185 parts.

The phenomena of liquid resemble those of gaseous diffusion. We have seen that hydrogen will descend to mix with chlorine, and that this heavy gas will ascend to diffuse itself among the particles of the hydrogen; so also, if a tall glass jar contain at the bottom one part of sulphuric acid, and over this two parts of the blue infusion of litmus, introduced so as not to mix with the acid, the heavier acid will in two or three days have diffused itself through the liquid, and the progress of the diffusion may be noted by the gradual change in colour from blue to red.

His

The subject of diffusion is greatly indebted to Professor Graham, who has obtained some interesting results by very simple means. apparatus consisted of a number of 4-oz. jars with openings in the a number of 30-oz. cylindrical vessels for containing the small jars. necks of 1.24 inch in diameter, for containing the trial solutions, and The solution to be tried was poured into a 4-oz. jar to within halfan-inch of the top; it was then filled up with pure water, closed with a glass plate, and let down into the cylindrical vessel, into which was carefully poured about 20 oz. of distilled water; the glass plate was then cautiously removed, and the apparatus was kept undisturbed at a steady temperature for several days. When the result had to be exwithdrawn from the vessel, the water of which was evaporated and the amined, the mouth of the solution jar was again closed and the jar From numerous experiments conducted in this way it was found: salt that had entered into it thus determined by weight. I. That when solutions of the same substance, but of different degrees of strength, were employed, the quantities diffused in equal times were cæteris paribus proportional to the quantity in the solution. Thus in 4 solutions of common salt, containing respectively 1, 2, 3, and 4 parts were, in No. 1, 2.78 grains, in No. 2, 5:54 grains, or just double; in of salt per cent., it was found that in 8 days the quantities diffused No. 3, 8:37 grains, or treble, and in No. 4, 11:11 grains, or four times the amount diffused from the first solution. II. That the quantities of the substances diffused from solutions containing equal weights Each solution contained 20 parts of the solid in of different bodies, varied with the nature of the substance, as in the following table. 100 parts of water, and was exposed for 8 days at a temperature of 60.5 degrees.

Weight in grains
diffused.
58.68
27.42
51.56
69.32
26.74
26.21
26.94
32.55
13.24

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Oil of vitriol
Sugar candy
Barley sugar
Starch sugar
Treacle of cane sugar
Gum arabic
Albumen

Saline substances were found to arrange themselves in groups, the
members of each group being equidiffusive, and the rates of diffusion
In No. 1, were
in each group connected with the rate of diffusion of the other groups
by a simple numerical relation. Isomorphous salts had generally equal
rates of diffusion. Seven such groups were made out.
hydrochloric, hydriodic, and hydrobromic acids, and perhaps also nitric
acid. These acids were found to be the most diffusible substances
known. Group 2, contains hydrate of potash and probably ammonia.
muriate of ammonia, and chlorate of potash. No. 4, nitrate of soda,
Group 3, nitrate of potash, nitrate of ammonia, chloride of potassium,
No. 5, sulphate of potash, carbonate of
and chloride of sodium.
potash, sulphate of ammonia, and ferro-cyanide of potassium; pro-
potash, and ferrid-cyanide of potassium. No. 6, sulphate and carbon-
bably also chromate and bi-chromate, bi-carbonate, and acetate of
ate of soda; and No. 7, sulphate of magnesia. In the following table

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III. Liquid diffusion is most regular in dilute solutions. It was also found that the quantity of a substance diffused from a solution of uniform strength increased with the temperature, but that the ratio of diffusion between different bodies at the same temperature was constant. It was found also, that of the whole quantity rather more than one-fourth was diffused during the first two days; the quantities diffused during each remaining period of two days being very nearly equal. IV. When two substances which do not combine chemically, but having different degrees of diffusibility, were mixed in solution and placed in a diffusion cell, the more diffusible substance passed out more rapidly than the other, so that a partial separation of the two bodies could be effected by this means. Thus the quantities of the carbonates of soda and of potash, diffused in the same time from a solution containing equal parts by weight, were as the numbers 35 and 65 nearly. It was even found, that in some cases chemical decomposition could be produced by liquid diffusion. Thus a solution of common alum (which is a compound of sulphate of alumina and sulphate of potash) arranged so as to diffuse into water, the sulphate of potash passed out more rapidly than the sulphate of alumina. V. It was found in the case of dilute solutions that one substance will diffuse into water which already contained another substance in solution, just as into pure water.

The reader who desires to pursue this subject further, is referred to Professor Graham's Papers, contained in the Philosophical Transactions' for 1850.

DIFLUAN. [URIC ACID.]

DIGAMMA, or VAU, is the name given by grammarians to a letter which once belonged to the ancient alphabet of the Greeks. It appears to have occupied the sixth place in that alphabet, for while epsilon is employed as the numerical symbol for fire, the next letter, as that alphabet is now arranged, is the representative of seven. Moreover, this position of the digamma will correspond precisely with that of rau or waf of the Hebrew, and of ƒ in the Latin alphabet, two letters of kindred power and form. A further argument may be found in the principles which would seem to have determined the arrangement of the Greek alphabet. [ALPHABET, vol. i., col. 235.] The letter is still to be seen in many inscriptions. [ALPHABET, vol. i, cols. 240-2, plate ii, Nos. 20, 21, 26.] With regard to the power of the letter, it is now the general and well-established opinion that it is equivalent to our own w. Its name has been evidently derived from the similarity of its symbol F to a repetition of the Greek gamma г. The use of the digamma prevailed more particularly in the Eolic dialect of the Greek tongue. In the other dialects it was commonly dropped, particularly the Attic; and as this became the favourite dialect of Grecian literature, the digamma at last escaped from the alphabet; and even the Homeric poems, which had been written in a dialect still possessing the digamma, were presented to the Athenians without that letter, to the serious injury of the metre. But though the form of the digamma was not admitted into the Attic alphabet, the vowel o was occasionally used, so as virtually to represent it, as in oida, oikos, orvos, equivalent to FIAA, FIKOZ, FINOZ (comp. the Latin video, ricus, vinum); and it was altogether superfluous to prefix the digamma, FOIKOZ, as was sometimes done. Other substitutions for the diganma are B, as Bowos for Fowos or oivos, eBaw for eFaw or eaw, Bpodov for Fpodov or podov, Вpiça for Fpiça or piça (Cf. Germ. wurz-el and our wort= root); secondly, ou, as Ovappav for Varro; thirdly, v, as auрηктоя for α-Γρηκτος οι αρρηκτος, αυως for afos or ηως. Whether is ever substituted is disputed. Some, supported by the analogy of the French habit, as gâter for vastare, and the repeated authority not only of the Hesychian glosses but of MSS., hold that yeder, yeiμata or ynμata, YETO are genuine dialectic varieties of Fe@ev, Feiuara, FEVTO. Ahrens on the other hand regards the F as a clerical error for F. A still more violent substitution is that of p for F, as dedрoikws for dedFolkws or dedouxws, of which we have the converse in our wubbish for rubbish. The Latin language, being more closely connected with the Eolic dialect of the Greek, is abundant in the use of this letter; for the true pronunciation of the v or u consonans must have been the same as our w, or it could not have so readily interchanged with the vowel u. The Greek words wor, eap, eσnepos, eoTia, tov, appear in Latin as ovum, vēr, vesperus, vesta, viola. In the last instance indeed there are two other points of difference, the Latin word being feminine, like rosa

ARTS AND SCI. DIV. VOL. III.

compared with the Greek podov; and secondly, a diminutive, which is well suited to the size of the flower. Sometimes a b appears in the Latin word, where the Æolic Greek must have had the digamma, as probus compared with rpaus; or an f, Formice compared with Ormiæ. The disappearance of the digamma in one dialect and its retention in another is in perfect accordance with what is seen in modern languages. In our own we have ceased to pronounce the w in who, whose, two, sword, answer, whole; while in one and once we have the sound without the character, and yet drop it again in only. The Danish dialect of the Teutonic language is remarkable for throwing off the w, thus word in the mouth of a Dane is ord.

For the assertions of the grammarians and the opinions of the learned with regard to the digamma, see Kidd's edition of Dawes's 'Miscellanea Critica,' pp. 175-335. The editor has given a list of the Greek words which he supposes once to have possessed this letter. more complete discussion of the subject is to be found in Ahrens, 'De Græcæ Linguæ Dialectis.'

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DIGEST. [CORPUS JURIS; JUSTINIAN'S LEGISLATION.] DIGESTER. A strong steam-tight vessel, usually of iron or copper in which water and other materials can be heated considerably beyond their boiling points; this form of apparatus was first contrived by Papin, and is hence frequently termed Papin's digester. The following is a description of two vessels of this description which are frequently employed by chemists.

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The first is constructed entirely of Low Moor wrought-iron, and consists of a cylinder ▲ ▲ (fig. 1), closed at the bottom, and welded in one piece by the steam-hammer. The cylinder is 18 inches long, inch Fig. 1. in thickness, and 3 inches internal diameter; it is furnished at top with a flanch, B B, 13 inch broad and inch thick, its upper surface turned true, and having an internal annulus sunk inch below the level of the surrounding surface. The cap of the digester, c c, is made to fit upon this flanch, with which it corresponds in thickness and diameter; it is furnished with a projecting face inch deep, fitting the mouth of the cylinder exactly. Within the circle of this projecting face, the cap is perforated by two apertures, into one of which is securely fixed the cast-iron tube d d, closed at the bottom, 6 inches long and inch internal diameter, forming a mercury bath for the reception of a thermometer. The other aperture, which is bouched with brass, serves as the bed of the safety valve e, which consists of a piece of brass wire inch diameter, slightly flattened on two sides and furnished with a head accurately ground to the surface of the cap: pressure is applied to this valve in the usual manner by the lever and weight fg. Both the flanch and cap are perforated by four holes for the reception of four screw bolts inch in diameter, which are inserted from below, and work into nuts that can be tightly screwed up by a lever key. The whole of the pressure produced by these screw bolts is made to take effect exclusively upon the surface of a leaden washer inch thick, placed in the sunken annulus above mentioned: and thus the apparatus is made perfectly impervious to gases and vapours, even under the enormous pressure of more than 100 atmospheres. In this digester, volatile liquids enclosed in glass tubes of large dimensions and moderate thickness of glass, may be exposed to any temperature below redness with safety. Water is generally used in this digester, but other and less volatile liquids may of course be substituted if desired: in most experiments, however, it is important that the pressure upon the exterior of the glass tubes should not be much less than that in their interior, and this condition is generally secured by the employment of water in the apparatus.

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The second digester is made of wrought copper and is of smaller dimensions, being especially designed for the heating of chemical substances without the intervention of glass tubes. It consists of a wrought copper tube, ▲ ▲ (jig. 2), 18 inches long, 14 inch internal diameter, and inch in thickness, drawn from a solid mass of the metal by a recently invented process. This tube is closed at bottom by a screw plug, and is furnished at top with a brass flanch, BB, 1 inch broad and 1 inch thick, screwed upon the copper tube: the vessel thus formed is closed by the brass cap c c, of the same dimensions as the flanch upon which it fits. The cap is furnished with a central projection 1 inch deep,

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fitting the copper tube, and pierced with a central aperture D, tapped to receive the screw plug E, which, with the intervention of a collar of lead, effectually closes the aperture. The cap c c is secured to the digester by three screw bolts inch in diameter, which are inserted from above and screw into the lower flanch. Perfect impermeability to gases is secured, as Fig. 3. in the iron digester, by a sunken annulus and a ring of lead. When it is desirable to collect the gases evolved during any operation in this digester, the plug E is replaced by a carefully made stopcock, to the nozzle of which a gas-delivery tube can be attached, when the reaction is completed. This digester is heated by means of a cylindrical copper oil-bath placed in a gas-stove, as shown in fig. 3. The gas stove consists of a strong wrought iron framework, A A, around which is fixed the sheetiron cylinder B B closed at the bottom, but furnished with draught regulator c, and contracted at top by a ring of sheet iron so as just to admit the cylindrical oilbath D D, the flanch of which rests upon the upper extremities of the wrought-iron framework, which are turned inwards for this purpose. The sheet-iron cylinder is surrounded by another of polished tin plate to prevent the too rapid radiation of heat; there is an interval of half an inch between the two cases, and both are pierced with holes at E for the exit of the products of combustion. A 4-inch copper pipe, F, pierced with eighteen or twenty small apertures, forms the gas-burner. By this arrangement, it is easy to maintain an almost constant degree of heat for any length of time. The temperature is ascertained by the thermometer f, immersed in the oilbath through an aperture bored in the cap and flanch of the digester for this purpose. This gas-stove is also used for heating the iron digester, but without the intervention of the oil-bath.

DIGESTION, in Chemistry, and especially in Pharmaceutic Chemistry, is the exposure of any substance to partial or total solution in a fluid, either at common temperatures or with a gentle heat: thus in the preparation of tinctures, the substance whose active principle is to be extracted is said to be digested in alcohol or spirit of wine, generally diluted with water. It is commonly performed in a glass flask or bolthead which should not be more than half filled, and covered with a piece of wet bladder, so that the evaporation of the menstruum or spirit may be prevented as much as possible; if the heat be so great as to endanger the bursting of the vessel or the bladder, the latter should be pierced with one or more small holes.

The flask may be heated either by means of the sun's rays, of a common fire, or of the sand-bath, or a stove: when the heat is so great that it would occasion the loss of a valuable menstruum, as spirit of wine, without any provision for condensing it being employed, a distilling apparatus should be made use of. Formerly a method of digestion called circulation was resorted to; this consisted in luting a close head on the digesting vessel, in which the vapour was condensed, and ran back into the digester without loss, it being condensed in the head merely by exposure to the air.

DIGIT, a finger, a term employed to signify any symbol of number from 0 to 9. According to the original application of the term, the first ten numbers should be called digits, but universal practice employs the word to signify the ten symbols used in reckoning number. Thus ten (10) is a number of two digits.

DIGITALIC ACID. A white crystalline acid possessing a peculiar odour, and found in purple foxglove. It is very soluble in water, less so in alcohol, and still less soluble in ether. It readily undergoes spontaneous decomposition, and has not been analysed.

DIGITALIN (CHO). This name has been given to the active principle of the purple foxglove; little is known of its properties, and it has by some been supposed to be a mixture of three bodies, called digitaline, digitalin, and digitalose. Further, digitalicrine, and digitalosine are also the names of bodies derived from the same source. The whole subject requires re-investigation.

DIGITA'LIS PURPU'REA, fox-glove, a biennial plant, of which the leaves and seeds are used. The leaves are sometimes accidentally confounded with those of different species of Verbascum, and of the Conyza squarrosa. The most powerful leaves are those procured from plants growing on the sunny sides of hills, those of the second year only should be collected. They must be carefully dried, and protected from damp, and kept in the dark. The active principle appears to reside in an extractive substance, which by careful evaporation may be

crystallised, and to which the name of Digitaline has been given. This principle is soluble in water, in alcohol, and sparingly in ether. It is very poisonous. One grain dissolved in a little water killed a rabbit in a very short time. Digitalis is given in powder, made into pills, or in an alcoholic tincture. Diversity of opinion exists respecting its primary action on the system, some writers considering it as primarily a stimulant, and the sedative effect a consequence of the previous excitement; others regarding it as a direct sedative. It is most likely that it acts in both ways according to the dose, and frequency of administration. If a small dose be given, and repeated at short intervals, a stimulant action will be most obvious, followed at a considerable interval by a sedative effect. If on the other hand a large dose be given at first, the sedative action is immediately displayed. The effect varies also with the position of the person, being different according as he is standing, sitting, or reclining.

Digitalis is the most perfect example known of a cumulative poison, as it may be used for some time, if the doses be small, without producing any manifest effect for several days, when suddenly faintness, intermittent pulse, giddiness, and other alarming symptoms appear. These are best combated by vital stimulants, such as warm brandy and water.

Digitalis has the power of reducing in a remarkable degree the heart's action, bringing down the pulse from 120 or more to 50 or 40 beats in a minute, and causing it to become at the same time intermittent. On this power depends its medical value in some diseases. It is remarkable, however, that while it thus lowers arterial action, it excites the absorbents and the kidneys to increased action, and so proves a valuable diuretic in dropsy and some other diseases. It is most useful in organic affections of the heart, and in the latter stages of some inflammatory affections, such as pneumonia, in phthisis pulmonalis, chronic peritonitis, and irritative or nervous fever. In the inflammatory or turgescent stage of hydrocephalus acutus, along with calomel, Golis states it to be very valuable.

As a diuretic, it is, like most medicines of that class, uncertain in its effects: it seldom answers if much inflammatory action exist when it is exhibited. To render it more certain it is generally given along with calomel, and squills, or some other diuretic. Digitaline has been recommended by Dr. Christison and others.

DIGITOLEIC ACID. A crystalline fatty acid, found in the purple foxglove. A number of salts have been prepared from it, but neither these nor the acid itself have been analysed. DIGNITIES. [TITLES OF HONOUR.] DI-IODOCODEINE. [OPIUM, ALKALOIDS OF].

DILAPIDATIONS. In its legal sense, this term is confined to the pulling down or destroying, in any manner, any of the houses or buildings belonging to an ecclesiastical benefice, or suffering them to run into ruin or decay, or wasting or destroying the woods of the Church, or suffering any wilful waste in or upon the inheritance of the Church. An incumbent is bound to keep the buildings in repair, restoring and rebuilding where necessary according to the original form, without addition or modern improvement, but he is not bound to supply or maintain anything in the nature of ornament, such as painting (unless necessary to preserve exposed timbers from decay), whitewashing, and papering. For damages calculated on this principle the successor may bring his action, either in the courts of common law or in the spiritual court, against his predecessor, or, if he be dead, against his executors.

The right to damages for dilapidations, as between other persons, is governed either by the laws relating to waste and repairs, or by special contract.

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DILATATION. [HEAT.]

DILATATION OF THE HEART. [HEART, DISEASES OF.] DILIGENCE, in the law of Scotland, is an expression nearly equivalent to "process" in the law of England. It includes the various means by which the person may be seized and imprisoned, the property attached and disposed of, to the end of enforcing payment of debt or performance of any civil obligation; and witnesses compelled to give their evidence, or to produce books, writings, &c. It would not give a comprehensive view of the object of this procedure to say that it is for the purpose of carrying out the judgments of the courts of law, because it is a characteristic of Scottish procedure that not only will an agreement properly executed between parties to allow diligence to proceed without a judgment (as in a warrant to allow judgment to be entered up in England) be a sufficient warrant for diligence, but there are certain pecuniary obligations, of which it is the characteristic that summary or the more rapid kind of diligence can proceed on them if they are not in any way vitiated or imperfect. These documents are bills of exchange and promissory notes, and the facility of recovering the debts contained in such documents, by a rapid method of execution, is a marked feature in the mercantile code of Scotland, and one which is believed to have beneficial effects. Before diligence can issue, such a document must be protested and registered in the books of some competent court, and thence the diligence issues, as if it were founded on the decision of the court. The registers being accessible to the public on payment of certain fees, this facility for tracing the initial steps of bankruptcy is made use of by members of the mercantile community of Scotland, who have a machinery by which they can inform each other of the persons whose names appear on the register.

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