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or unfixed. The longitudinal vibrations in a rod are produced by fixing it at one of its points, and by rubbing it in the direction of its length with a piece of cloth wetted, or powdered with rosin. Yet, in the latter case, we do not produce a sound, unless the fixed point of the rod be at its half, its third, its fourth, and, in short, a certain aliquot part. Analysis proves that the number of the transversal vibrations of rods and laminæ of the same nature is in the direct ratio of their thickness, and in the inverse ratio of the square of their length. The breadth of the lamina has no effect on the number of their vibrations; it only causes the force which produces the vibrations to vary in its intensity. In elastic rods of the same nature, the number of longitudinal vibrations is in the inverse ratio of their length, whatever may be their diameter and the form of their transversal section. In fig. 137 is a representaFig. 137.

considered preferable, we fix it at some point in its surface, and, through a hole pierced in the middle, we produce friction by means of horse-hair powdered with rosin.

Vibrating plates present nodal lines which vary in number and position, according to their form, their elasticity, the

Fig. 138.

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tion of an instrument, the construction of which is founded on the longitudinal vibrations of rods.

This instrument, constructed by M. Marloye, consists of a solid wooden stand, on which are fixed twenty cylindrical fir rods, some coloured and some white. Their lengths are determined in such a manner, that the white rods give the diatonic scale, whilst the coloured rods give the semitones and complete the scale, rendering it chromatic. In order to play an air with this instrument, the rods are rubbed in the direction of their length, between the thumb and the fore-finger, which have been previously dipped in powdered resin. The sounds which are thus obtained have a strong resemblance to those of the Pandion reed.

mode of putting them in action, and the number of their vibrations. The nodal lines are rendered obvious to the sight by covering the plates with a thin stratum of sand before putting them into the vibratory state. As soon as the vibrations commence, the sand leaves the vibrating parts and disposes itself along the nodal lines as shown in the preceding figure.

The vibrations of plates are subject to the following general laws: In plates of the same nature and of the same form, giving the same figures, the number of vibrations is in the direct ratio of their thickness and in the inverse ratio of their surface. Any particular note will always produce the same figure with the same plate; but a small change may be produced in the figure by slightly changing the place at which the plate is held, without causing any difference in the tone. If the tone be changed, the existing figure disappears at once, and a new one makes its appearance. The lowest note which any plate yields produces the simplest figure; and the higher the note is the more complex the figure, or, in other words, the more nodal lines there will be. If similar plates of various sizes be made to vibrate in the same manner, similar figures will be produced in each. The notes, however, will differ; the larger plates will yield the lower notes; and under equal dimensions, the thicker or stronger plates will yield the higher

notes.

Vibrations of Membranes.-The flexibility of membranes prevents them from vibrating unless they are stretched like the top of a drum. Then they yield a sound which is higher in proportion to the smallness of their dimensions and the force with which they are stretched. M. Savart constructed vibrating membranes, by pasting very flexible goldbeater's skin on wooden frames. Membranes are made to vibrate by percussion, as in the drum; or by the influence of other vibrating bodies. Thus, M. Savart has observed that a membrane can be put into a vibratory state by the influence of the vibrations of the air, whatever may be the number of these vibr tions, provided that they are sufficiently intense. Fig. 13 represents a membrane vibrating under the influence of vibra Vibration of Plates.-When we wish to put a plate into a tions impressed upon the air by a sonorous bell. Some fine state of vibration, we fix it at the middle point, as represented sand spread over the membrane shows the formation of swells in fig. 138, and we draw a bow across its edges; or, as may be and nodes in it, in the same manner as in plates. Fig. 139,

LESSONS IN CHEMISTRY.-No. XXV. THE apparatus for burning a mixture of coal-gas and atmospheric air, as represented and described in the preceding lesson, may be considered as the representative of a wind-furnace on a small scale. By slightly modifying, in a manner now to be described, it admits of being changed into the representative of a blast-furnace. To this end, the apparatus, as already deFig. 16.

of rough work out of it. For example, he will use it as the source of heat for keeping up a supply of distilled water. This is the plan I adopt in my own laboratory, and as its description will serve to initiate the matter of distillation, displaying that process in its simplest form, I shall describe it somewhat in detail.

Fig. 17.

scribed in the preceding lesson, is supplied with a central jet, perforating the wire-gauze and communicating with a flexible tube. To the latter a mouthpiece being adapted, the breath can be directed in a current upwards through the flame, thus concentrating the energies of the latter against one point, after the manner of a blowpipe, fig. 16. By means of a blast of this description, gold and silver admit of ready fusion, provided the crucible holding them be of hot too great dimensions.

Although the mixed gas flame is that which answers the greatest variety of purposes, nevertheless other descriptions of flame are occasionally advantageous. Of these, the flame resulting from a circular burner with small lateral orifices is amongst the most useful. One great advantage it possesses is the following: instead of heating the lowest point of a flask or retort, its energies are rather directed on the sides; thus lessening the depth of fluid through which the bubbles resulting from ebullition have to pass, and thus diminishing the chances of fracture.

The Argand Gas Burner.-As a specific source of heat, the Argand gas burner is not so much employed at this time as formerly; nevertheless it affords the best means of using gas for the purposes of illumination, and will therefore be found in many laboratories. A prudent chemist will never allow the heat from such a source to go to waste; he will get some sort

First of all, what do we mean by distillation? Most persons associate with this process some intricate operation, involving the use, as a necessity, of much apparatus; but it is not necessarily of this kind. Viewed in its simplest aspect, distillation only differs from evaporation in this,-that whilst in the latter case the vaporised product goes to waste, in the former case it is condensed and preserved; such, at least, is true as regards the distillation of liquids and solids; but the liberation of gases from certain substances by the application of heat is also termed distillation. The apparatus employed in my laboratory for the distillation of water is as follows. The source of heat, I have already mentioned, is an Argand burner; over this is placed a permanent loop of strong wire, securely fixed to the wall in such a manner that a convenient support for a large glass flask results, as indicated in the accompanying diagram fig. 17. This support, before use, is wound round with a little tow, in such a manner that the glass flask does not directly bear upon the iron. The tow rapidly chars, but the carbonaceous residue

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remains still, affording a soft cushion for the flask, which holds about three quarts, to rest upon.

To the mouth of this flask is loosely adapted a cork, to which a length of glass tubing, about the eighth of an inch in diameter (no more), is attached by perforation. This tube is attached to another by means of an india-rubber connecter b, and tube after tube is then attached, until a length of about 18 feet results, fig. 18. This total length is suspended by cord loops from the ceiling, and finally terminates, as represented, in a large wide-mouthed glass jar c, passing loosely through a cover of tinned wood . This cover, it will be seen, is supplied with a second perforation, admitting the bent glass tubes, which is joined by means of an india-rubber tube i to another glass tube p, the latter represented in our diagram as looking upward, and secured in that position by a wire hook h. Perhaps it is scarcely necessary to indicate that the tube passing from the flask to the receiving jar must have a gradual fall; this fall need, however, only be of the slightest.

Turn we now to the apparatus in action. The flask being removed from its stand and charged with water (about threeparts full), is replaced and the cork inserted, a very slight amount of pressure being sufficient and the rotation of the cork unnecessary. The gas may now be lighted, taking care that the flame is small for the first minute or two; the gas may then be turned on fully, and the apparatus left to itself. Ebullition soon ensues and vapour is eliminated; this vapour passing along the tube, gets partially condensed into water, which being driven onward by the force of uncondensed steam, both enter the receiving iar together. Here the greater part of the remaining steam is condensed, not enough escaping to produce inconvenience, nor, indeed, to be for the most part appreciable. When it is desired to withdraw a portion of water from the receiving jar, this is accomplished by merely unhooking the tube P, bending it on its india-rubber joint i, and bringing it to the position of the dotted line s'. This arrangement being made, a syphon will result and water will flow; the delivery being perfectly at command, and ceasing so soon as the tube pis bent back upon s. The apparatus just described illustrates the process of distillation under its most simple form. The operation of cooling, or condensation, the student will observe, is left pretty much to itself; and so long as water is the liquid to be distilled, the contrivance answers perfectly; but liquids whose volatility is greater than water require a different treatment when distilled. Sometimes artificial means must be had recourse to for condensing them, and hence the necessity for instruments presently to be described. On the large scale, in various operations of manufacturing chemistry, the usual plan of accomplishing distillation is by means of a still and worm, as the respective parts of the arrangement are termed. The still is subject to considerable variations, but in general terms it may be described as a vessel of copper or other metal, formed on the type of a flask, to which a head and delivery-tube are attached, usually by means of cement or lute; a small model of this kind is represented in the accompanying diagram, fig. 19.

Fig. 19.

Attached to this instrument is its necessary complement-a worm or refrigerator, consisting of a metallic pipe spirally

arranged inside a vessel holding water. The operation of this worm is almost too obvious for comment. The heat of vaporisation being first communicated to the metal, the latter imparts it to the water, which in its turn becomes gradually so hot that it has to be renewed, and the condensed vapour passes through the lower extremity of the worm into a receiving vessel. The stili and worm, though not a bad contrivance for large commercial operations, may be said to be obsolete in laboratories for analysis and general research; flasks and retorts, properly arranged, and connected with instruments of condensation, taking their place.

A retort, I believe, is always associated with the idea of distillation in the amateur chemist's mind; but on trial he will be surprised to find to what an extent this comparatively expensive instrument may be discarded in favour of flasks on the large scale and tubes on the small. There are certain liquids, of course, which do not admit of being distilled in flasks. Mineral acids are of this kind; their vapour being so corrosive, that the cork where with the mouth of the flask is necessarily closed up being rapidly destroyed; thus not only disarranging the apparatus, but contaminating the result; but even in many cases of this kind the operation may be conducted in flasks, provided india-rubber stoppers be made to take the place of corks. In general terms it may be stated, that the success of distillation does not so much depend on the kind of evaporating vessel, as on the perfection of the means employed in bringing about condensation. The chief contrivances to this end will be described in our next lesson.

LESSONS IN ALGEBRA.-No. XIII.

(Continued from p. 345.)

INVOLUTION, OR RAISING OF POWERS.

172. When a number is composed of the product of the same factor any number of times, the result is called a power of the factor. Powers are divided into different orders or degrees; as the first, second, third, fourth, fifth powers, &c., which are also called the root, square, cube, biquadrate, &c.

The powers take their names from the number of times the root, or first power, is used as a factor in producing the given power. The original quantity is called the first power or root of all the other powers, because they are all derived from it. Thus, if 2 be the root or first power, then

2 X 2 = 4, the square or second power of 2. 2 X 2 X 2 = 8, the cube or third power. 2 X 2 X 2 X 2 = 16, the biquadrate or fourth power, &c. And, if a be the root or first power, then

a Xa aa, the second power of a. axaxa aaa, the third power. axaxaxa

aaaa, the fourth power, &c.

173. The number of times a quantity is employed as a factor to produce the given power is generally indicated by a figure or letter placed above it on the right hand. This figure or letter is called the index or exponent, Thus a X aaa, is written a2 instead of aa; and a X a Xaaaa, is written a3.

The index of the first power is 1; but this is commonly omitted, that is, ala.

An index is totally different from a co-efficient. The latter shows how many times a quantity is taken as a part of a whole; the former how many times the quantity is taken as a factor. Thus 4aa+a+a+a; but at a Xa Xa Xaaaaa. If a 4, then 4a 16; and at 256,

174. Powers are also divided into direct and reciprocal. Direct Powers are those which have positive indices, as d2, ds, &c., and are produced by multiplying a quantity by itself, as above described. Thus d x d = d2; d x d x d = d3; and dxdxdx d= d1.

The Reciprocal Power of a quantity is the quotient arising from dividing a unit by the direct power of that quantity, as

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19

1

d2 ds d

&c.

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1

before the index; thus - =d-, &c. The direct and reci

procal powers of d, are d1, d3, d2, d1, d', d − 1, d - 2, d - 3, d4, &c. in which do 1.

16. INVOLUTION is the process of finding any power of a quantity as explained in Art. 172.

177. To involve a quantity to any required power.

RULE. Multiply the quantity by itself, and by its successive products, till it is taken as a factor as many times as there are units in the index of the power to which the quantity is to be raised.

All powers of unity or 1 are the same, viz. 1. For 1 X 1 X 1 X 1, &c. 1.

178. A single letter is involved or raised to any power, by giving it the index of the proposed power; or by repeating it as a factor as many times as there are units in that index. If the letter or quantity has a co-efficient, it must be raised to the required power by actual multiplication.

EXAMPLES.

repeated n times,
Ans. 27.x3.
Ans. 256y'.
Ans. 128a7.

1. The 4th power of a, is at, or aaai. 2. The 6th power of y, is y, or yyyyyy. 3. The nth power of 2, is a", or xxx ... 4. Required the 3rd power of 3x. 5. Required the 4th power of 4y. 6. Required the 7th power of 2a. 179. The method of involving a quantity which consists of several factors, depends on the principle, that the power of the product of several factors is equal to the product of their powers.

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25. Find the 2nd, 3rd, and nth powers of

1 1 a2

180. When the root is positive, all its powers are positive also; but when the root is negative, the ODD powers are negative, while the EVEN powers are positive.

26. Find the cube of

27. Find the nth power of

Ans.

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28. Find the square of

aym
a3× (d+m)
(x + 1)3

anymn

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a2 + ab
+ ab + b2

(a + b)1 = a3 +2ab+b2,

the first power

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a+b
a2+2ab+ab
+ab+2ab2 + b3

(a + b)3 = a + 3a2b + 3ab2 + b3,
atb

a2 + 3a3b+3a2b2 + ab3

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+ a3b+3a2b2 + 3ab3 + b1

(a + b) 1 = a1 + 4a3b + 6a2b2 + 4ab+b, the fourth 2ab+b2.

power.

30. Find the square of a-b. Ans, a2 — 31. Find the cube of a +1. Ans, a3 +3a2 + 3a + 1. 32. Find the square of a+b+h. Ans. a2 + 2ab+ 2ah + b2+2bh + h2.

33. Required the square of +2d + 3. Ans. a2 + 4ad + 6a +4d2 + 12d +9. Ans, b' +863 +246

34. Required the 4th power of b + 2.

Hence any odd power has the same sign as its root. But an even power is positive, whether its root is positive or nega+ 166 +16. tive. Thus (+ a) × (+ a) = a. And (-a) × (— a) = a2. 181. To involve a quantity which is already a power. RULE.-Multiply the index of the quantity by the index of the power to which it is to be raised.

EXAMPLES.

aaaaaɑ

14. Find the 3rd power of a2. Here, (a2)3ab,
For aaa; and the cube of aa is aa X aa aa
a; which is the 6th power of a, but the third power of a2.
15. Find the 4th power of ab. Ans, al.
16. Find the 3rd power of 4ar. Ans. 64a3.

17. Find the 4th power of 2a3 × 3x2d. Ans. 1296a12“d1.
18. Find the 5th power of (a + b)2.
19. Find the 2nd power of (a + b)".
20. Find the nth power of (xy).

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35. Required the 5th power of x + 1.
+x2+x+1.

36. Required the 6th power of 1
20631566b3 + b3.

Ans. af 203

- b. Ans. 1 66 + 156

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Here it will be seen, that in each case, the first and last terms are the squares of a and h; and that the middle term is twice the product of a by h. Hence the squares of binomial and residual quantities, without multiplying each of the terms separately, may be found by the following rule :-

(1.) The square of a BINOMIAL, the terms of which are both positive, is equal to the squares of the first and last terms, plus twice the product of the two terms.

(2.) The square of a RESIDUAL quantity is equal to the squares of the first and last terms, minus twice the product of the two terms. EXAMPLES.

39. Find the square of 2a + b. Ans. 4a2 + b2 + 4ab. 40. Find the square of +1. Ans. 2 + 1+ 2h. 41. Find the square of ab+cd. Ans. a2 b2 + c2 d2+2abcd. 42. Find the square of 6 y +3. Ans. 36 y2 + 9 + 36y. 43. Find the square of 3d-h. Ans. 9 d2 + h2 — 6dh. 44. Find the square of a — - 1. Ans. a2 + 1 — 2a. 185. For many purposes it will be sufficient to express the powers of compound quantities by exponents without an actual multiplication.

EXAMPLES.

45. Find the square of a + b. Ans. (a+b).

46. Find the nth power of be + 8+x. Ans. (be +8 + x)n. In cases of this kind, all the terms of which the compound quantity consists must be included in the parenthesis.

186. But if the root consists of several factors, the parenthesis used in expressing the power, may either extend over the whole, or may be applied to each of the factors separately, as convenience may require.

Thus the square of (a + b) × (e + d), is either

{(a + b) × (c + α) } 2, or (a + b)2 × (e + d)2. The first of these expressions s the square of the product of the two factors, and the last is the product of their squares, and these are equal to each other.

In like manner the cube of a × (b+d), is {a × (b + d)}3,

or a3 × (b + d)3.

187. When a quantity, whose power has been expressed by a parenthesis, with an index, is afterwards involved by an actual multiplication of the terms, it is said to be expanded. Thus (a+b), when expanded, becomes a2 + 2ab+b', and (a+b+ h) becomes a2+2ab + 2ah + b2 + 2bh + h2.

BINOMIAL THEOREM.

188. To involve a binomial to a high power by actual multiplication is a long and tedious process. A much easier and more expeditious way to obtain the required power, is by means of what is called the BINOMIAL THEOREM. This ingenious and beautiful method was invented by SIR ISAAC NEWTON, and was deemed of so great importance to mathematical investigation, that it was inscribed on his monument in Westminster Abbey.

To illustrate this theorem, let the pupil involve the binomial a+b, and the residual a b, to the 2nd, 3rd, and 4th powers. Thus, (a + b)2 = a2 + 2ab + 62.

(a + b) 3: = a3 +3a3b+3ab2 + b3.
(a + b) 1 = a1 + 4a3b + 6a2b2 + 4ab3 + b1.

Also (a

b)2 = a2

(a - b)3 = a3 (a — b) 1 = a1

-

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2ab+b2. 3a-b+3ab2

· 4a3b+6a2b2

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By a careful inspection of the several parts of the preceding operation, the following particulars will be observed to be applicable to each power, especially if carried out to a greater number of powers.

1. By counting the terms, it will be found that their number in each power, is greater by 1 than the index of that power; thus, in the 3rd power the number of terms is 4; in the 4th power, it is 5, and so on.

2. If we examine the signs, we shall perceive, when both terms of the binomial are positive, that all the signs in every power are+; but when the quantity is a residual, all the odd terms, reckoning from the left, have the sign+, and all the oven terms have the sign Thus in the 4th power, the signs

of the first, third, and fifth terms are +, while those of the second and fourth are —.

3. As to the indices, it will be seen that the index of the first term, or the leading quantity* in each power, always begins with the index of the proposed power, and decreases by 1 in each successive term towards the right, till we come to the last term from which the letter itself is excluded. Thus in (a + b)a the indices of the leading quantity a, are 4, 3, 2, 1.

4. The index of the following quantity begins with 1 in the second term, and increases regularly by 1 to the last term, whose index, like that of the first, is the index of the required power, Thus in ja+b) the indices of the following quantity 5, are 1, 2, 3, 4.

5. We also perceive, that the sum of the indices is the same in each term of any given power; and this sum is equal to the index of that power. Thus the sum of the indices in each of terms of the 4th power is 4.

6. As to the co-efficients of the several terms, that of the first and last terms in each power is 1; the co-efficient of the second and next to the last terms, is the index of the required power. Thus, in the 3rd power, the co-cfficient of the second and next to the last terms, is 3; and in the same terms in the 4th power, it is 4, &c.

It is to be observed, also, that the co-efficients increase in a regular manner through the first half of the terms, and then decrease at the same rate through the last half. Thus, in the 4th power they are 1, 4, 6, 4, 1,

in the 6th power they are 1, 6, 15, 20, 15, 6, 1. 7. The co-efficients of any two terms equally distant from the extremes are equal to each other. Thus, in the 4th power, the second co-efficient from each extreme is 4; in the 6th power, the second co-efficient from each extreme is 6; and the third is 15.

8. The sum of all the co-efficients in each power is equal to the number 2 raised to that power. Thus, (2) 16; also, 64; so the sum of the co-efficients in the 6th power, is 64. the sum of the co-efficients in the 4th power, is 16, and (2)" :

189. If we involve any other binomial, or residual, to any

required power whatever, we shall find the foregoing principles

true in all cases, and applicable to all examples. Hence, we may safely conclude, that they are universal principles, and may be employed in raising all binomials to any required power. They are the basis or elements of what is called the Binomial Theorem.

The BINOMIAL THEOREM my be, therefore, defined, a general method of involving binomial quantities to any proposed power. It is comprised in the following general rule:

I. SIGNS.-If both terms of the binomial have the sign +, all the signs in every power will be +; but if the given quantity is a residual, all the odd terms in each power, reckoning from the left, will have the sign+, and the even terms -

II. INDICES. The INDEX of the first term or leading quantity, must always be the index of the required power; and this decreases regularly by 1 through the other terms. The index of the following quantity begins with 1 in the second term, and increases regularly by through the others.

III. CO-EFFICIENTS.- -The co-efficient of the first term is 1; that of the second is equal to the index of the power; and, universally, if the co-efficient of any term be multiplied by the index of the leading quantity in that term, and divided by the index of the following quantity increased by 1, it will give the co-efficient of the succeeding

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