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is mixed with powdered charcoal, and made into a paste with 300 parts of white sand (silica), 100 of carbonate of soda, cilor starch; the mass is subdivided into pellets; these are placed 43 of slaked lime, and 300 of cullet, or broken glass, of the in a tube of porcelain, which is heated in a charcoal furnace, same kind. These ingredients are placed in a conical crucible and a current of chlorine is passed through it. The aluminum and melted. The crucible is allowed to stand some time in chloride condenses in a cool receiver. The chemical action is a high temperature, in order that any impurities may settle thus expressed

to the bottom. The fused glass is then poured out on a table A1,0, + 3C + 6C1 = Al,CI, + 3C0.

of cast iron, and the thickness is regulated by ledges of the

same metal. The liquid glass is inade to cover this table Aluminum Sulphate (A1,380,) is made for the use of the rapidiy by a heavy roller passing over it. dyer in large quantities in the north of England. Clay, which As is the case with all glass, the plate is next annealed, has been roasted, is acted upon by half its weight of sulphuric that is, it is allowed to cool very slowly in an oven whose scid. At Whitby, however, this addition of acid is unnecessary. temperature is gradually diminished during a fortnight. If The alnm schist contains iron pyrites and in the slow roasting glass be cooled at all suddenly, it becomes so brittle as to be some of the sulphur of this compound becomes sulphuric acid. useless. The grinding and polishing of the plate is effected by Water is added, and the clear liquid is drawn off; the iron it causing one plate to move upon the surface of another by contains is precipitated as Prussian blue by sodium ferro- machinery, the grinding materials being fine sand and water. cyanide. On evaporation, thin flexible scales of the salt are When a smooth surface is thus obtained, the plate is polished, obtained.

first by fine emery and then by peroxide of iron, Alums are double salts formed by aluminum sulphate and an Bohemian glass is a silicate of potash and lime, and on account alkaline sulphate; thus common alum is Al,K,450, + 24H,0, of its infusibility is used in the laboratory, that is, it is a compound of A1,350, and K,SO (potassium sul Bottle glass, the colour of which is immaterial, is composed of phate) with twenty-four molecules of water of crystallisation. 100 parts of sand, 80 of soaper's waste, 80 of gas lime, 5 of It is found native in volcanic neighbourhoods, where the sul clay, and 3 of rock salt. phuric acid from the volcano has combined with the potash and Flint glass has these ingredients :-Pure sand, 100; red lead, alumina in lava to produce it. At the Whitby Alum Works 20; pearl-ash, 40; nitre, 2; and cullet, 100. the iron is not precipitated as Prussian blue. In the roasting it The lead gives great lustre to the glass, by increasing its becomes green vitriol (sulphate of iron). To the clear liquid, refractive power; hence this glass is used for optical purposes. which contains aluminum sulphate and iron sulphate, a solution In its manufacture the crucibles are closed, lest the air should of potassium chloride is added. The iron sulphate now becomes "reduce" the lead. It will be frequently observed that this iron chloride, and the potassium sulphate with the aluminum takes place in a piece of tubing, if held in the blow-pipe sulphate form alum. This is separated by crystallisation, and flame for some time. It at first appears as if the tube was sent to market.

“smoked.” Soda Alum, or Ammonia Alum, may be made by using the Coloured glasses are obtained by the addition of various salts of those alkalies, instead of the potassium salt, in the above metallic oxides. Gems are imitated by making a glass which process.

contains as much as 53 per cent. of lead oxide. This is termed The sesquioxides of iron, chrome, and manganese are capable of "paste." taking the place of alumina, with its compounds; hence they are Porcelain differs from glass in the great preponderance of silisaid to be isomorphous with alumina. When they do this in cate of alumina in its composition. It consists mainly of clay, alam, iron alum, chrome alum, or manganese alum is produced. which is infusible, and some alkaline silicate which fuses and Clay is aluminum silicate, and is the result of the decomposition binds the clay together, rendering it impervious to moisture. The of felspar, one of the constituents of the primary rocks. When fineness of the ware entirely depends on the purity of the clays, clay is got directly from the felspar, it is called kaolin, or etc., from which it is made. The glaze which covers porcelain is porcelain clay.

produced by dipping the " biscuit" ware into water in which is Clay, when it absorbs water, becomes soft and plastic, offering suspended finely-ground felspar ; the porous mass absorbs the the best of mediums by which the tender rootlets of plants can moisture, leaving the surface covered uniformly with the felspar. take up the potash, ammonia, etc., necessary for their growth. It is then exposed in seggars to a very high temperature, by which The clay used for making bricks must have a large proportion the felspar is melted and the glaze produced. Stone ware and of silica in its composition.

common "pottery ware" are glazed by means of common salt. The following table will at once show the difference of the The ware is dipped in sand and water, placed in the furnace, various clays :

into which has been thrown moist salt. The heat quickly con

verts the salt into vapour. In the presence of the steam the Cornish Stourbridge

Brick
Kaolin.
Pipe-clay. Blue-clay.

silica on the ware decomposes the salt, forming a silicate of Fire-clay.

clay.

sodium which glazes the erticle, and hydrochloric acid which 46-32 64.10

49.44

escapes. Alamina 39-74

38:04

The other nine metals of this class having no interest, we pass Iron Oxide. 0.27

774 to the "metals proper." Lime.

1.48 Magnesia 0.44

trace Potash

SYMBOL, Zn - COMBINING WEIGHT, 65 --SPECIFIC GRAVITY, 7:1 Sodas 12.67

MELTING POINT, 423° CENT.
Water

10.00
13.57

This metal is never found native, but is associated with

Silica.

46.38

34.26

23.15
1.58

53.66
32.00
1.35
0.40
trace

1.04
1.20

0-36

ZINC.

0-95

5.14

12.08

1.94

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sulphur and carbonic acid in its two chief ores, blende anl 99.80 100-05 99.49

100.00

calamine. The former is zinc sulphide, and in England is Alumina enters into the composition of numerous minerals.

usually found with galena, the sulphide of lead, in mountain The zeolites contain their water of crystallisation, and there- limestone; the latter is zinc carbonate, of which there are

mines in the Mendip Hills. fore swell up before the blow-pipe-hence their name. Analcime, natrolite, and stilbite are examples of this class. Granite, basalt, crushed, is roasted; by this process the sulphur of the blende is

Extraction of the Metal from its Ores.--The ore, after being garnet, mica, and lazulite are familiar aluminous minerals.

burnt away as SO,, and the carbonic acid of the calamine is GLASS, PORCELAIN, ETC.

driven off. Glass is a compound of silicates of the alkalies and of the The oxide of zinc which remains is mixed with half its weight of alkaline earths; its properties are too well known to require powdered coal, and placed in large clay crucibles; these are heated description. There are various kinds of glasses which have in a furnace, their delivery tubes passing through holes in the certain distinctive properties due to their composition which walls. At first the blue flame of carbonic oxide issues from render them valuable for specific purposes.

them, and when this changes to white the metal is distilling. A Window glass consists of silicates of soda and lime. Soda long iron pipe, eight feet long, is then fitted on the discharge produces a more fusible glass than potash; and lime increases pipe, and zinc distils into iron vessels placed to receive it at its lustre and hardness. The best manufactory of plate glass the end of this tube. is at St. Gobain, in France, where these ingredients are used: Zinc is a very crystalline metal. A little below 150° it is soft

100.23

and ductile, but at 200° it becomes so brittle as to bear pound To complete my remarks on the alphabet, I must now say ing in a mortar. At a bright red heat it melts and volatilises, something of the letters K, W, X, and y, important letters in and its vapour barns with a bright yellowish flame into the English, but which do not occur in Italian. oxide.

Instead of k, the Italians use before consonants, and before Zinc can decompose both steam and carbonic acid at a red the vowels a, o, and u, the letter c; and before the vowels e heat, and, as we have seen, is dissolved in sulphuric, nitric, and and i, ch. For example, instead of Kalend, the Italians write hydrochloric acids, giving off hydrogen and forming the corre- Calende. sponding salt. With nitric acid, however, the action is varied The English letter w does not occur at all in Italian. with the strength of the acid and the temperature ; nitric oxide, The letter X, which represents, properly speaking, a compound nitrous oxide, nitrogen, or ammonia being given off according to sound (ks), is unknown in pure Italian words, and the English circumstances.

sound is never heard. In words of foreign origin, which would Zino precipitates all metals less oxidisable than itself from have this sound in English, the Italians place an s or ss, or c; their solutions; but a strong boiling solution of potash dissolves as for the word example (from the Latin exemplum), the Italians it, hydrogen being liberated and an oxide of zinc formed, write essempio; for extreme (from Latin extremus), they write which is dissolved by the alkali. When exposed to the air, it estremo; for Xenophon, Senofonte; for Xerxes, Serse ; for Alex. becomes covered by a firmly-adhering coat of oxide, which pro- ander, Alessandro. The letter c replaces the x in words which tects the metal beneath from any further action of the air; are the compounds of the prefix ex, when c follows it; for hence it is used to galvanise iron, which is effected by plunging example, for excellent they write eccellente; for excess, eccesso

, a sheet of iron, perfectly clean, into a bath of molten zinc, etc. Custom has, however, sanctioned the use of the w in a few covered by sal-ammoniac; the zinc readily adheres to the iron. words of Greek origin, for Xantippe and Xanto (Xanthus, the

It forms many useful alloys. Brass is the most important, river in Asia Minor) are just so written in Italian. They are consisting of two parts of copper and one of zinc. German nevertheless pronounced as if they were written Santippe and Silver is brass whitened by the addition of a little nickel. Santo. (The latter word has retained the x principally that it

Zinc Sulphate, or White Vitriol (ZnSO.), appears, when a might not be confounded in writing with the word Santo, a solution of zinc in sulphuric acid is evaporated, as colourless saint.) four-sided acicular prisms.

The letter y is always replaced in Italian by i; as, for ex. Zinc Sulphide (Zn) is precipitated as a white gelatinous ample, for physics (physical science), the Italians say fisica; for mass when sulphuretted hydrogen passes through a solution of stygian, stigio. zinc sulphate. The presence of a mineral acid prevents the

SECOND PRONOUNCING TABLE, precipitation.

Showing the combination of Vowels with Semi-Vowels in Zinc Chloride (ZnCl.) is largely used as an antiseptic, as

Natural Order. “Burnett's Disinfecting Fluid.” It is easily obtained by dissolving the metal in hydrochloric acid.

That my pupil readers may thoroughly exercise themselves in The salts of zino are distinguished by giving no precipitate pronunciation, in order to give a complete illustration of the with sulphuretted hydrogen in acid solutions. They yield a junction of vowels and semi-vowels in natural order, I have white precipitate, with potash, soda, or ammonia, which is selected words of two syllables, in which the first syllable of soluble in excess of the alkali.

the first word is the same as the concluding syllable of the If placed on charcoal in the blow-pipe flame, when moistened second. with cobalt nitrate, & green residue remains which is not Italian. Pronounced.

English. fusible.

Fere
fê-rai

Beasts.
Refe
rái-fai

Thread.
Foce
fô-tchai

Mouth (of a river).
LESSONS IN ITALIAN.-IV.

Cefo
tchè-fo

A monkey.
III.-THE SEMI-VOWELS (continued).

Frigo
fóo-go

I put to flight.
Gufo
góo-fo

A horned owl.
I HAVE not yet spoken of the letter H. It is named in the
alphabet acca (pronounced ah'k-kah). According to its alpha-

The vowel u in Italian, as a final letter, is only to be found betical sound, and because its two syllables are substantially in monosyllables ; as, tu, thou; fu, was ; or in those words that one, only placed inversely, it might be classed as a semi-vowel; have the grave accent on the last syllable; as virtù, virtue ; but as it is only an auxiliary letter to modify the sounds of Corfù, Corfu. I am therefore compelled, by the use of the word and g, as I shall have occasion to explain fully hereafter, it is a gufo, and others to follow, to depart from the strict system. mere soundless, written sign, not a letter. It also serves to dis Lago

láh-go

Lake. tinguish the words ho, I have, from o, or; hai, thou hast, from Gola

gó-lah

Throat. ai, dative plural of the article; ha, he has, from a, the preposi.

laí-zo

Hurt. tion"to"; and hanno, they have, from anno, the year. This dis

Sole
só-lai

Sun.
lée-tchai

It is permitted. tinction is, however, only for the eye, for, in pronouncing, the h

Cieli
tchè-lee

The heavens. is quite mute ; and some purists, headed by Metastasio, instead

Lode
lô-dai

Praise. of an h, put the grave accent in those first four words.

Delo
dê-lo

Delus.
The Italian has no aspirates, which essentially distinguishes

Lume
lóo-mai

Light. it from the leading languages of Europe. Only in the middle,

móo-lai

Mules. and at the end of some few interjections, a kind of aspiration Maro

máh-ro

Wild basil. is heard, which is only produced by the prolongation of the

ró-mah

Rome. sound of the vowel, or of the transition of the voice from one

mái-zai

Month.

Seme vowel to another-principally, however, by a more emphatic

sái-mai

Seed.

mée-rah emotion by which such interjections are thrown out; as, for

The sight in artillery, aim, Rami ráb-mee

Branches. example, ah! ahi! deh! ahimè ! eh! oh! ehi! ohi! ohime! doh!

Modo mô-do

Manner, mode. In the early period of the language, the Italians wrote all

dô-mo

Tamed. words manifestly of Latin origin with an initial h; as, for Muro

mbo-ro

Wall. example, habile, now abile; hinno, now inno; hora, now ora;

róo-mo

I reconsider. historia, now istoria. This insignificance of the h has given Navo

náh-vai

Ship. rise to some proverbial expressions: as, “Questa cosa non vale Vena

vai-nah

Vein. un'acca,” “this is not worth an h,” or, as an Englishman would

Lái-rah

Negress. ráh-pai

Frogs. say, “not worth a fig, or a farthing ;” or, “ Non m' importa un'

née-tchai

Berenice, a woman's name. acca," " I don't care an h for it," or, as an Englishman would

Ceni
tchái-nee

Thou suppest. say, “I don't care a straw for it;” or “ Non ne saper un' acca,"

Nome
nó- mai

Name. " not to know an h of something;" or, as is often said in Eng.

Meno
mái-no

Less. land, "an iota of it." When an Italian has to pronounce the Nuca

nóo-kah

Nape of the neck. h in another language, it is only with the greatest difficulty he Cuna

kóo-nah

Cradle. can master it.

Rado
ráh-do

Thin, rare.

Leso

Lice

Mule

Roma
Mese

Mira

Domo

Rumo

Nera
Rane
Nice

Soma

Tom.

Dora
đó-nh
He gilds.

with “ duels of honour," in which men have met each other in Ross rái-zai Surrenders (cf towns).

the field, and risked dear life itself, for the supposed preservaSero sê-rai Sir.

tion of their name from insult, or their courage from question. Rido rée-do I laugh.

But the world itself has long since decided tisat such a use of Dori do-ree

Thou gildest. Roba

the word honour is neither legitimate nor wise-as under many 6-bah

Property, victuals, merchandise,
Baro
báh-ro
A cheat.

[robe.

circumstances it may be more honourable and heroic even to Rudo rio-dai Rude.

endure injustice and wrong, than to resent it at the precions Dure dóo-rai Durations.

sacrifice of human life. It was a fact that oftentimes the most Sera sáh-rah Sarah.

dishonourable men could cleverly handle the rapier, or point Rasa ráh-zah Erased.

the pistol; and victory was seen to be a question of nicelyScoo sai-ko With himself,

balanced skill, rather than of well-contested principles. Honour Cose ko-sai Things.

is something far nobler than any mere exercise of physical This is the plural of cosa, thing (pronounced kô-sah), one of courage such as the duel embodied; it is more connected with those exceptional words where the s must be pronounced with a that moral courage of which in these articles we have had sharp, hissing sound, though it is placed between two vowels. to speak—a courage which knows how to endure as well This exception should be imprinted on the reader's memory, as how to act. No outward possessions can make up for the because, as is obvious from its meaning, the word is of the most absence of this moral quality; those who wilfully forfeit honour frequent occurrence.

may be well assured that as no wealth can buy it, so no material Sire sée-rai Formerly Sir, now Sire.

grandeur or estate can compensate for its loss. If, indeed, it is Reso rái-zo Rendered.

felt by us that an amassed fortune has been accumulated by sô-mah Burden.

oppression or wrong, there will always be a lessened respect Naso máh-zo

awakened in us concerning the possessor; and although our eyes Befare bef-fáh-rai To scoff.

may be dazzled by his wealth, our moral sense will be affronted Ofeso of-fái-zo Offended.

and pained too, if it be understood that the loud-voiced panegyrist Soffice sóf-fee-tchai Soft, flexible, surple.

of virtue does not pay his own debts or neglects his own 'family Soffogo sóf-fo-go

I suffocate. Sususo

--then his public professions will be, as Young says, " but the soof-fóo-zo

Wetted. Corallo ko-rahl-lo Coral.

gibbet of his name." Vitello vee-tel-lo Calf.

The sense of honour should be cultivated in children. NaCavillo kah-vil-lo I annoy, quibble.

turally prone to selfishness, it is well to show them that the Satollo sah-tól-lo

Satisfied, satiated, tired. mere gratification of self will lead to that love of ease and Catullo kah-tóol-lo Catullus.

indulgence which would rather do the pleasant than dare Cenammo tehai-náhm-ino We supped.

the right; and in the outgrowth of early character it is of Dilemma dee-lemn-inah Dilemma (logical).

the highest moment to learn that we may lose more by one Enimma ai-nim-mah Enigma.

[chin.

mean or ignoble act than we can gain by a multitude of hard Sommomno som-mom-mo A blow with the fist under the

endeavours.
Affummo abf-foom-mo I smoke (meat).
Inganno in-gáhn-no
Deceit.

The abbreviations Hon. and Right Hon. in the English lanAntenna an-tên-nah Yard (of a ship).

guage show us what a value is set upon that honour of which Erinni Ri-'n-nee The Furies.

these titles are the symbols; and abroad, to be ranked amongst dr.no ah-ron-mi Aaron.

the Legion of Honour is the highest dignity that the French ah-lon-no Alumnus, pupil.

nation can bestow. True, indeed, we may never be called upon Caparm kah-páhr-r.... Earnest money.

to draw a sword or to defend our country, but none can watch Atterto alt-térr-ro I knock down.

the course of commerce without feeling that noble posts are open Butirro boo-tárr-ro Butter,

to honourable men. It is evident that in all circles in the end Ricorro ree-kórr-ro I recur.

character tells. Noble deeds are blazoned forth by the "coat of Azzurro ah-dzóor-ro

Azure. Abbasso ahb-bábs-80 Low.

arms” which is all unconsciously being painted by every good Orseseo OS-Sē8-60 Possessed, bored, dunned.

man's life; and as honour is not a matter which appertains Afisso ahf-fís-so Affixed.

merely to noble lineage or to high estate, it may be a common Indosso in-dôs-so Upon the back.

distinction attainable in every sphere of human life. To be disConcusso con-kóos-80 Moved, shaken, contrite. honourable is in the end to be despised ; not that ignoble acts

always show themselves at first; there may be some consider

able period before the seed sown shows its up-springing rank ESSAYS ON LIFE AND DUTY.-XIV. grass, but the dishonourable life will by slow but steady pro

gress become known, and then will ensue loss of character and HONOUR,

of reputation too. THE definition of the word honour in olden times was this-a The proverbs of nations show that the question of honour is noble kind of seignory or lordship; and, inasmuch as honour one of the most potent matters in all climes. Thus there is an is the crown of character, we may regard its possession as one English proverb against the dishonour of meanness—"The of the patents of a true nobility. It is generally understood to groat is ill-saved which shames its master;" and the Spanish mean nobleness of mind-magnanimity-a scorn of meanness, one against hasting to be rich careless of principle—_"Quien springing not only from fear of reproach, but from an innate en un año quiere ser rico, al medio le ahorcan" (He who will be loyalty to principle. Honour came in a secondary sense to rich in a year, at the half-year they hang him); and the German mean reputation, or a good name; and it sometimes happened one showing that dishonour even in little things affects larger that men were more sensitive concerning reputation, or what affairs—"Ungerechter pfennig verzehrt gerechten thaler" (The Fas thonght or said of them, than about character, which was unrighteous penny corrupts the righteous pound). Thus we see, what they really were. We take honour, in its primary sense, in these proverbial crystals of truth, how dishonoar is supposed to mean true nobleness of mind--a lofty scorn for the base, the to bring with it loss and disgrace. We might, as a kind of per low, the shabby, the tricky, the false, the unjust, and the mean; contra, show the favourable estimate of honour which has been and most assuredly in the formation of character this is one of also condensed into national proverbs. Thus there is the proverb the most to be coveted possessions. We are told, on the highest “ La verdad es hija de Dios" (The truth is daughter of God), authority, that a good name is rather to be chosen than great picturing forth, as it does, the dignity and immortality of honourriches; and certainly to be dishonoured, as it is one of the able character ; as also the Swiss proverb, "It takes a good severest punishments, is also one of the saddest disgraces of many shovelfuls of earth to bury the truth ;" showing that there life.

is a beantiful resurrection for all honourable characters, even It may be admitted that the word honour has been greatly though they may have been unjustly defamed. We may be misused. It has come to be associated with debts of honour, sure of this, that judged by the ultimate outcome of things, to or debts which conld not be sued for in a regular way in our be dishonourable does not pay. It may, for the time being, common law courts, but which it is supposed no honourable man seem to serve the purposes of aggrandisement and success ; but would escape from on that account; and it is also associated it is only for a brief and uncertain period; the fabric of am.

bition, built with untempered mortar, soon falls to the ground. factors are to be multiplied, the product will be the same in The desire to be honoured is a righteous aspiration. To seek the whatever order the operation is performed. public prize, or the public post, is to enter the arena of life with 74. If the multiplicand be a compound quantity, each of its a determination to win the esteem of men, and the rewards which terms must be multiplied into the multiplier. Thus the product are incident to honourable toil. Every one must feel a thrill of of b+c+d into a, is ab + ac + ad. For the whole of the pleasure when the hard working Grecian passes to his college multiplicand is to be taken as many times as there are units in career, and when the humble peasant succeeds in founding a the multiplier. family or a name. These rewards of honour no man is for

EXAMPLES bidden to seek. Honour, however, has not always its material (1.) Multiply d + 2xy (2.) 2h + m (3.) 3h +1 (4.) 2hm +3 rewards. Honour may, indeed, be gained by their relinquish- By 35

6dy
my

46 ment. In the heroism of duty circumstances may require their abnegation, and the honour of the brave spirit may be its own Product: 3bd + 6bæy 12dhy+6dmy 3hlmy + my 8thm +126 best reward. Righteousness and truth are, however, evidently at 75. It must be carefully observed that the preceding instances the basis of this world's government; and it is manifest to all are not to be confounded with those in which several factors are who have eyes to discern the secret of success, that the honour connected by the sign X, or by a point. In the latter case, the able man is as much on the high road to outward advancement multiplier is to be written before the other factors without being as he is in the straight path to inward happiness and self-repeated. The product of 6 x d into a, is ab x d, and not respect.

ab x ad; for bx d is bd, and this into a is abd (Art. 70).

The expression b x d is not to be considered like b + d, a con. LESSONS IN ALGEBRA.-V.

pound quantity consisting of two terms. Different terms are MULTIPLICATION.

always separated by + or [Art. 19). The product of

6 xh x m x y into a, is a xbxh x m Xy, or abhmy. But 66. EXAMPLES.—(1.) What will 4 oranges cost at æ pence 6th + m + y into a is ab + ak tam + ay. each?

76. If both the factors are compound quantities, each term in Here we say, if one o-ange costs x pence, 4 oranges will cost the multiplier must be multiplied into each term in the multipli 4 times as much ; they will therefore cost 4x pence; and this is cand. Thus (a + b) into (c + d) is ac + ad + bc + bd. For the the answer.

units in the multiplier a + b, are equal to the units in a, added (2.) How much can a man earn in 5 months at a pounds per to the units in 6. Therefore the product produced by a must month? Reasoning as before, we have a X 5 = 5a pounds for be added to the product produced by b. Whence, the produet the answer.

of c + d into a + b, is ac + ad + bc + bd. Now 4x is equal to x +*+*+-&; and 5a = a ta ta ta ta.

For the product of c+d into a is ac + ad; and the product 67. Hence the repeated additim of a quantity to itself is called of c + d into b is bc + bd [Art. 75]; therefore the product of MULTIPLICATION. From this definition of multiplication, it is manifest that the product is a quantity of the same kind as the c+ d into a + b is ac + ad + bc+ba.

EXAMPLES . multiplicand. 68. It is plain, therefore, that multiplying by a whole number

(1.) Multiply 3x + d

(2.) 4ay + 25 is toking the multiplicand as many times as there are units in the

Ву
2a + hm

3c + 97 multiplier. Thus multiplying a by 1, is taking the multiplicand Product: bax+ 2nd + 3hme +dhm 12acy + 6bc + darwy +2hara once, as a.

(3.) Multiply
a +1

(4.) 2b -+7 Multiplying a by 2, is taking the multiplicand trrice, as

By
3a + 4

60 +1 a + a, etc.

69. On the other hand, multiplying by a FRACTION is taking a Product: 3ax + 3x + 4a + 4 12bd + 420 + 25 + 7, certain PORTION of the multiplicand as many times as there are

(5.) Multiply d + reth by 6m + 4 + 7y. Ans. 6dm + 6mm like portions of a unit in the multiplier. Thus:

+ Chm + 40 + 4rx + 4h + 7dy + 7rxy + 7hy. Multiplying a by , is taking of the multiplicand once,

(6.) Multiply 7 + 6 + ad by 3r + 4 + 2h. Ans. 21r + 18 as fa.

+ 3adr + 28 +245 + 4ad + 14h + 125h + 2adh. Multiplying a byl, is taking of the multiplicand tuice, as

77. When several terms in the product are alike, it will b Ja + .

70. Multiplying two or more letters together, is writing them expedient to set one under the other, and then unite them by th one after the other, either with or without the sijn of multipli- rules for reduction in addition, as in the following examp.es:cation between them [seo Art. 23, page 21]. Thus b multiplied (1.) Multiply b + a

(2.) b +0+2 into c is 1 x c, or b. c, or bc; and the product of a into y, into By

bta

b+c+ 3 >, is a X y X z, or .Y.£, or, as it is more commonly written,

bb + ab

bb + bc + 25 dyz. Also the product of am into æy is amzy; and of abc into

+ ab + ao

+ bc + cc + 2c rys, is abcxyz.

Product: 6b +- 2ab + ao

+ 35 + 3c+ 6 71. There will be no difference as to the result in whatsoever order the letters are arranged. Thus the product of ba is the

bb + 2bc + 5b + cc + 50+ rame as that of ab; and 3 times 5 is equal to 5 times 3. In

(3.) Multiply a+ y +1 like manner, the product of a, b, and c, is abc, cab, bac, or cba.

Ву 36 + 2x + 7 It is more convenient, however, to place the letters in alpha

3ab + 3by + 36 betical order. 72. When the letters have numerical CO-EFFICIENTS, these

+ 2ax + 2y + 2 must be multiplied together, and prefixed to the product of the

+ 7 + 7y +7 letters.

Product: 3ab + 3by +36 + 2ax + 2xy + 2x + 7 + 7y EXAMPLES.-(1.) Multiply sa into 2).

(4.) Multiply 3a + d + 4 by 2a + 32 +1. Anus. 6a+ 11 Here the answer is bab. For if a into b is ab, then 3 times a

+11a + 3d2 + 13d + 4. into b is evidently 3ab ; and if, instead of multiplying by b, we

(5.) Multiply b + cd + 2 by 3b + 4cd + 7. Ans. 362 +7 multiply by twice b, the product must bo twice as great, that is + 136 +422d2 + 15cd + 14. 2 X 3ab, which is bab.

(6.) Multiply 36 + 24 +h by a Xdx2x. Ans. 6abur +40 (2.) Multiply 12hy (3.) 3dh (4.) 2ad (5.) 76dh (6.) 3ay + 2adha, Ву 2ra my 13glum

8mx 78. It is plain that when the multiplier and multiplic Product: 24hrxy3dhmy 26adghm 7bdha 24amæy

consist of any quantity, repeated as a factor, this factor will

repeated in the product as many times as it is in the multir 73. If either of the factors consist of figures only, theso must and multiplicand together. bo multiplied into the co-efficient of the other fretor, and the

EXAMPLE.-Multiply e xa xa letters annexed. Thus 3ab into 4, is 12ab; 36 into 2x, is 72x;

Ву аха and 24 into hy, is 24hy.

From the preceding rules we have the general onc, that when Product: a Xaxa xa xa = aaaaa, or as.

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Here a is repeated three times as a factor in the multiplicand, little moment provided enough power is gained to perform the and twice in the multiplier ; hence, it is repeated five times in required work. the product, and is called the fifth power of a.

We now pass on to the second class, called breast wheels (Fig. EXAMPLES.-(1.) What is the product of bbbb by bbb? Ans. 25). These have usually an advantage over the class we have just bbbbbbb, or b?.

spoken of, though they are inferior in power to overshot wheels. (2.) What is the product of aa X aaa x aaaa by aaa x aaaa? The stream here, Ans. aaaaaaaaaaaaaaaa, or alo.

instead of being al79. It is also plain, from Art. 73, that the numeral coefficients lowed to run in its of several factors should be brought together and made into natural bed, is stopone factor by multiplication. Thus to multiply 2a x 36 by ped by a dam or 4a x 56, gives the product of 2a X 36 X 4a x 56, or 120aabb; embankment built For the co-efficients are factors (Art. 24), and it is immaterial in across it. The water what order these are arranged. Therefore 2a X 36 X 4a x 5b accumulates behind 2 x 3 x 4 x 5 X 6 X 4 x 6 x 6=120aabb.

this till it rises to EXAMPLES.-(1.) What is the product of 3x x 4x x 5y by the level of the top, 2y 43 ? Ans. 480xxyyz.

and then flows over (2.) What is the product of 3a x 4bh by 5m x 6y? Ans. at the same rate as 360abhmy.

before. A curved

Fig. 25. (3.) What is the product of 45 X 6d by 26 +1? Ans. channel is made for 481 de + 24bd.

it, in which the wheel is placed, with its floats nearly touch80. The product of two or more powers of the same quantity is ing the sides, so that little water can escape. Much thereexpressed by writing that quantity with an index equal to the sum fore depends upon the accuracy of the workmanship; if the of the indices of the proposed powers. Thus the product of a2 water leaks by between the floats and the brickwork, there and a® is as; and the continual product of 23, ah, and x is æ13. is a corresponding loss of power ; while, on the other hand, So likewise the product of æm and an is am + , and that of a and if the floats scrape against the sides, there is a loss by friction. nu is 2.0 ++; and, on the same principle, the product of qm _ . In this wheel the floats are frequently curved or bent in the and zo is cm. The reason of this is evident from Art. 79. Thus middle, so as to hold the water better. It is important to take a and as are the same as aa and aaa; the product of which is care that the water, after it has left the wheel, flows away anana or a”; the index 5 being the sum of the indices 2 and 3, without obstructing its motion, as a great loss of power somethe numbers which show how often a is used as a factor in the times results from this not being done. A step is frequently given powers.

put in the course, as shown in the illustration, so that, as soon EXAMPLES.—(1.) What is the product of a’ and as? Ans. a?. as the water has reached the lowest point of the wheel, it falls (2.) Find the continued product of a?, ab, and a62? Ans. a?b3.out of contact with it. Where this cannot well be done, part

(3.) Find the continued product of 2-3, x?y, zdys, and æy. of the water is allowed to run by a channel at the side of the Ans. 3:22y.

wheel, and by its momentum produces a current which aids in carrying away the tail-water.

In the overshot wheel, which is represented in Fig. 26, the KEY TO EXERCISES IN LESSONS IN ALGEBRA.

water is usually diverted from the stream and conducted by a EXERCISE 5.

series of troughs to the top of the wheel, where it pours into 1. 2ab + 4zy + 10dfg. 15. x + 2bbb.

the buckets. The end of the trough is sometimes open, so that - 2522 - 6ab-7m. 16. 5yy + 2 laga.

the water flows on with the impulse it has acquired; the better 3. - Say - 31b-9bc.

17. 16 (x + y) – 10 (a + b). plan, however, is for it to be closed, and an opening made un4. 2ab - 16xy - 4d.

18. - 13 (a + b) – 52(x - y). demeath, as shown in the figure, through which the water may 5. lla + 3x + 48f + 18xyz. 19. a + 2aa + 7ba.

flow. 6. 1760 - 42xy + 23gh.

11xxx.

In this way it falls on the wheel without any momentum, 7. 9108 + y + ac - ay 4a + bc 21. 17 + 172.

and all strain is avoided. With this wheel nearly all the - 5+ ye + de.

22. - 7a - 333. 8. 213 + 40xy

power of the water may be employed, as it may be made of such 13a + 5b6 Joab 23. 5«y – 5ax + ay – my – 52 + 7df.a width that none of the stream runs to waste. It is found - 42. 24. 87a.

best to let the wheel turn very slowly, as thus there is very 9. Gry -- 29ab.

25. 4roza - Q® + be. 10. 1Say - Par. 26. 8x'y + 8x18.

little momentum left in the water when it leaves the wheel, and 11. a + b +c+d-f+g-h- xy. 27. a3 – 44* + 11as.

all loss from the water being thrown off by centrifugal force is 12. 13ab + 4xy - bad. 28. 2a.

avoided. Cogs are, therefore, usually placed round one edge of 13. - (a-b+c+d-f- gh). 29, 203 + a2x + 2ar? - 23.

the wheel; these work in a pinion, and the motion is imparted 14 - (- ab + cda - af + x + y 30, 6x® + 2.

to the machinery from this pinion instead of from the axis of the - ghf + bc – xys). 31. (a - c) ? + (6 + d) y.

wheel. The shape of the
buckets is, perhaps, of more

importance in this than in LESSONS IN HYDROSTATICS.-VI. the other descriptions of BREAST WHEELS-OVERSHOT WHEELS—TURBINES—BARKER'S

wheel, for, since the weight

of the water is the moving MILL-HYDRAULIC RAM-MACHINES FOR RAISING WATER -PERSIAN WHEEL-ARCHIMEDIAN SCREW.

power, it is important to

retain as much of it as pos. We examined in our last lesson the principle, and saw some of sible in the buckets until the advantages and disadvantages, of the undershot water they arrive at the lowest Wheel. There is one other advantage, however, which should point. Sometimes they are be just noticed, and that is that when made in its simplest made of the shape shown form, with the floats radiating directly from it, it can be used in the figure, and masonry in a tidal stream.

erected to confine the water, In any other kind of wheel, the water must flow in one uni- as in the case of the breast

Fig. 26. form direction. The level should also be nearly even, as the wheel, but more frequently height at which the water strikes the wheel makes a great the buckets are made to curve upwards, and thus retain the difference in its working. With this kind, however, we have water. merely to fix the wheel between moored barges, so that it may Another matter has, however, to be considered in deciding on rise and fall with the tide, and we shall have it almost con- the best shape of bucket, and that is to allow the air to escape. stantly at work, part of the time turning in one direction, and as the stream pours in it has to displace the air, and with some part in the other. Only a small portion of the force of the shapes of bucket this opposes and scatters the water in a much stream will, it is true, be utilised; but, as there is usually in greater degree than would at first be expected. Openings are tidal rivers very much more power than is needed, this is of sometimes made in the cylinder for this purpose. In the best

20. x - 2x

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