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ταχυς, Swift

OF THE KEYS AND THEIR SIGNATURES.

LESSONS IN GREEK.-No. XVI. You are always to suppose that the staff is in the key of C, unless some sign is placed at the beginning

BY JOHN R. BEARD, D.D. which points to another key note. Hence the Gi key of c is called the natnral key (although it is

*

EXERCISES.GREEK-ENGLISH. not really more natural to the ear or voice than any other), and the other keys in use are de- * E * UNDER the name of adverbs we indicate those indeclinable veloped from this. The diagram at the side

words which denote the relations of time and place, or the represents the key of c, with its "semitones” # Di * relations of way and manner; as Exel, there; vuv, now; kalws, between the third and fourth, and seventh and

well. eighth. If we take the fifth of that key (6),

* C * Adverbs of manner are formed from adjectives, by affixing and wish to raise another key upon it, the dia- B wg to the pure stem of the adjective. As a practical rule you gram will show you that we shall require a new

may take this note, instead of F, and a “chromatic semitone

A above it; in fact, the TU of "transition." In

The Genitive Plural of the Adjective is changed into ws, b.y. order, then, to adapt the staff to the key of G, a * G * mark' like a double cross, called a “sharp,' is

Adj.
Gen. Plural.

Adverb.
placed on P, at the beginning. It means that all * F
the r's on the staff are raised to suit the key of

pilos, loving φιλων φιλως, lovingly E

καλος, beautiful καλων καλως, beautifully If, again, we take the fifth of that key D, for a D

απλούς, simple απλών απλώς, simply key note, it will only cost you the drawing of

πας,
all

παντως, altogether another diagram to prove that we shall not only C

σωφρων, wise

σωφρονων σωφρονως, wisely need the P sharp, but also another sharp upon c.

ταχεων

ταχεως, swiftly EXERCISE 20. Develop by diagrams four other keys ascending

μέγας, great μεγαλων μεγαλως, greatly by fifths. Remember that in reekoning musical intervals you

αληθης, true αληθων αληθως, truly include the two extreme notes.

συνηθης, accustomed συνηθων συνηθως, according to If now we take the fourth of the c key (or F) for a new key

custom, pote, the diagram will show you that we shall want a new note instead of B, a chromatic semitone lower, in fact the FI of “tran- The terminations Oev, O., and 8€ form adverbs by being sition.” In order then to adapt the staff to the key of F, a mark added to nouns, pronouns, and verbs, to signify relations called a flat is placed upon B at the beginning. It makes all of place; thus bev denotes, from a place (whence), O., at a the B's on the staff “flat.

place (where), and de, to a place (whither): e.g. ovpavodev,

from heaven; ovpavob., in heaven; ovpavovde, to heaven. With EXERCISE 21. Develop by diagrams four other “flat keys" (pronouns de becomes ge, thus alloge, to some other place; 80 ascending by fourths.

with ekel, there, as EKELOE, thither. In the plural of the substantives in ας, σδε passes into ζε, as Αθηναζε for Αθηναστε; from Aonvai, wv, the city Athens.

Adverbs of place terminate in w, as avw, above; karw, below; etw, without ; eow, within. There are many adverbs which are obviously cases of nouns or pronouns, as carins, (80 in Latin, derepente) suddenly; fov, somewhere; Otrov, où, where ; autov, there ; ovdajov, nowhere ; these adverbs are all genitives.

Accusatives are also common, as a pwny, at the davon ; parpain

a long way; nepav, beyond the river, whence the country along Bb Eb ab

Db

the east side of the river Jordan had the name of Perea, that is, the other side : dwpeav, gratis, gratuitously; onuepov, to-day (Lat. hodie); avplov, to-morrow (Lat. Cras).

G

D

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C.

Comparison of Adverbs. These “flats" or “sharps "at the beginning of the staff are Adverbs of manner have commonly no peculiar adverbial called the “signature" of the tune. Their only use to the termination, but employ, in the comparative, the neuter sinsinger is to point out the key note.

gular, and, in the superlative, the neuter plural of the corresTO FIND THE KEY NOTE, therefore, remember that the last sharp that the neuter singular of comparatives may be used adverb

ponding adjectives. The same fact may be stated thus, namely, towards the right hand stands upon TE (TE, the “piercing note, will easily associate in the memory with sharps), and that nou ially, that ;, with an adverbial signification; and that the is consequently the next above. Remember also that the last neuter plural of superlatives may be used with an adverbial flat towards the right hand stands upon fal (associate “flat

signification; e.g. with “ desolate note") and that now is the fourth below.

from

S. EXERCISE 22. Put the proper key signatures to all the preced-coows (σοφος), wisely σοφώτερον σοφωτατα ing exercises.

σαφως (σαφης), clearly σαφεστερον σαφεστατα EXERCISE 23. Write from memory the signatures of the keys xapevtws (xapuis),charmingly xapuotepov χαριέστατα G, D, A, E, F, 2 flat, and a flat. These are the keys most used. To evda povus (evdaywv), happily evðaWoveotepov evda poveotara remember these signatures, notice the place of the first sharp aloxpus (αισχρος),shamefullyαισχιον

αισχιστα and of thy first flat. Then the sharps descend a fourth, ascend ηδεως (ηδες), pleasantly ήδιον

ήδιστα a fisth, and so on; while the flat signatures ascend a fourth,

(raxus), swiftly θάττον descend a fifth, and so on.

ταχιστα Thus they necessarily fall into Taxews parallel rowe. Verify these remarks, and they will greatly help

Adverbs of place in w retain that termination in the comyour memory.

parative and superlative. The note tu is expressed in the old notation by a sharp before the note which would otherwise have been faw, except in

8. ανω-κατω tunes with flat signatures, when & "natural" is used instead.

κατω τερω

C. ανω τερώ

avw, above Katw, below

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περα, beyond

εγγυς, near

πη, whither :

TOTE, then

που, where : πως, how :

τως,

The comparative and superlative of most other adverbs of gods; TEpitaons, es, suffering greatly, acutely sensible to sufer. place end in w, as

ing ; θηριωδης, ες, like the animals, Iow, douncast.
C. περαιτέρω
S. (none)

5. αχαριστος, η, ον, ungrateful ; αμελης, ες, neglectful;

πλεονεκτης, αναricious ; απιστος, unfaithful. τηλου, at a distance τηλοτερω

τηλοτατω έκας, at a distance έκαστερω

έκαστατω

6. εκτος, adν. without; τα εκτ. αγ. external goods, advantages ;

ανωφελης, ες, useless και χωρις, αραrt from ; τοις εχουσι, to those εγγυτέρω

εγγυτατω who have (them), that is, their possessors. Some adverbs have a reciprocal relation to each other. The 8. ws, how, somehow, some way, in a measure ; the adverb simple forms stand as relatives. By prefixing a to the restricts or qualifies the statement. relatives, you make direct interrogatives. Put o before the π, and you convert the direct into indirect interrogatives, and we should say, D. and P., have two sons.

9. Δαρειου και Παρυς. These genitives depend on παιδες; indirect relatives. Prefix r instead of #, and then you obtain demonstratives ; as

10. φιλοπενθης, ες, fond of mourming; πενθος, ους, τo, grief,

lamentation. Simple Rel. Direct Interrog. Indirect I. & R. Demonstrat. 11. to . The infinitive mood with the article is often ή, whither

όπη
ty, there, thither equivalent to a noun in English, to injure another is worse

than to suffer an injury.
ηνικα, when
πηνικα, at what

οπηνικα τηνικα, at that time?

time 12. ο μεγ. βασιλ. the great King, that is, the king of Persia, όθεν, whence ποθεν, whence ? οποθεν τοθεν, thence

who was the great king to the Greeks ; EKELVOS, he, that one. oi, whither ποι, whither ? όποι

.13. όρων, seeing, pres. part. from οραω; επι τω, &c., in oneOTE, when

quence of having many disciples ; xopos, our chorus, here class, όποτε ποτε, when?

audience και συμφωνος, ον, agreeing, harmonious ; ο εμος, mine ; ού, where

όπου

literally, the mine. ώς, as

όπως

14. ανδρ. παντ. σοφωτ. &c. The superlative governs a Of these forms oi, ty, to ev, and tws, are found only in the genitive; thus we say in English “the fairest of women." poets, and consequently are not to be ordinarily used in prose 15. φυσικως, naturally, by natural impulse; λειος, α, ον, soft, composition.

mild, sweet και χρηστικος, η, ον, useful.

18. εποιησας (from ποιειν) ελιou hast done και προειπων, in foreEXERCISES PROM THE CLASSICS.-GREEK-ENGLISH.

telling (that). 1. Παν το σκληρον χαλεπως μαλαττεται.

2. Ο ουρανός

19. τους αλισκ, the captives ; αλισκειν, to take, capture ; χαλκους εστι τα εξω. 3. Ελεγεν ο Βιας, ατυχη ειναι τον ατυχιαν κλεπτειν, to roo; τιμωρεομαι, I punish. μη φεροντα. 4. Η φιλοσοφια διδασκει, ότι δει μητ' εν ταις 20. εσται, it will be, future of ειναι. ευπραγιαις περιχαρείς υπαρχειν, μητ' εν ταις οργαις περι

21. τιμφη, could he honour ? τιμαω, I honour. παθεις και θηριωδεις. 5. Πώς η αχαριστοι, η αμελεις, πλεονεκται, η απιστοι, η ακρατεις ανθρωποι δυνανται φιλοι

22. μειρακιον, a young man; ανδραποδον, ου, το, a slave. γιγνεσθαι ; 6. Ο πλουτος και τα εκτος αγαθα χωρις αρετης

ENGLISH-GREEK. ανωφελη εισι τοις εχουσι. 7. Τίς ορνις ευφωνoτερα εστιν αηδονος ; 8. Αι δευτεραι πως φροντιδες σοφώτεραι. 9. Δαρειου

Wise men seek not external advantages. Women suier και Παρυσατιδος γιγνονται παιδες δυο πρεσβυτερος μεν Αρταξ. very much in adversity. An intemperate man cannot become

a faithful friend. The nightingale is the sweetest (in voice) ερξης, νεωτερος δε Κυρος. 10. Γυναικες ανδρων φιλοπενθεστεροι | of birds. Girls are more given to sorrow than women. The 11. Το αδικειν κακιον εστι του αδικεισθαι.

12. “o wisest (man) is greatest. I am admired for having much Αγησιλαος περι του μεγαλου βασιλεως ειπεν. Ti yap epov wealth. How can men admire me for having much wealth? μειζων εκεινος, ει μη και δικαιοτερος : 13. Ζηνων όρων τον | My brother is wise, my father is wiser, the philosopher is

wisest. θεοφραστον επι τω πολλους έχειν μαθητας θαυμαζομενον, ο

Children naturally. love their parents. Fight, O

citizens, well and bravely for (περι) your (the) city. εκεινου μεν χορος, εφη, μειζων, ο εμος δε συμφωνοτερος. 14. Σοφος Σοφοκλης, Ευριπίδης σοφώτερος, ανδρων δε παντων Σωκράτης σοφωτατος. 15. Η μελιττα φυσικως εν

Certes, j'ai eu souvent dépit de voir des juges attirer, par fraude ανθεσι εξανευρισκει το λειοτατον μελι και χρηστικωτατον. 16. et fausges esperances de faveur ou pardon, le criminela découvrir Παντες, α επιστανται, ραστα τε και ταχιστα και καλλιστα και εon fait, et y employer la piperie et l'impudence. Il servirait bien ήδιστα εργαζονται. 17. Καλως και ανδρειως έκαστα ποιει. 18.

à la justice, et à Platon même qui favorise cet usage, de me fournir Ο Αστυάγης, καλως, εφη, εποιησας, προειπων. 19. Τους αλισ- ne Pestime pas moins blessée par soi-meme que par autrui.

d'autres moyens plus selon moi: 'est une justice malicieuse ; et κομενους, ως κακως κλεπτοντας τιμωρουνται. 20. Αδηλον εστι | Montaigne. ειτε βελτιον ειτε κακιον εσται, 21. Πώς καλλίον η ευσεβεστερον On est quelquefois un sot avec de l'esprit, on ne l'est jamais avec τιμφη θεους; 22. Σωκρατης ιδων μειράκιον

πλουσιον και

du jugement.-La Rochefoucauld. απαιδευτον, ιδου, εφη, χρυσουν ανδραπαδον.

D'où vient qu'un boiteux ne nous irrite pas, et qu'un esprit

boiteux nous irrite ? C'est à cause qu'un boiteux reconnaît que VOCABULARY.

nous allons droit, et qu'un esprit boiteux dit que c'est nous qui boitons; sans cela nous en aurions plus de pitié que de colère.

Pascal. 1. σκληρος, α, ον, dry, λαrd; παν το. The article is frequently used in Greek when it must be omitted in English, as

Ceux qui jugent d'un ouvrage par régle, sont, à l'égard des in general propositions και μαλαττομαι, I am softened.

autres, comme ceux qui ont une montre, à l'égard de ceux qui n'en 2. ta etw, in regard to the things without, that is, on its L'autre dit : il n'y a que trois quarts d'heure. Je regards ma

ont point. L'un dit: il y a deux heures que nous sommes ici. exterior.

montre ; je dis à l'un : vous vous ennuyez; et à l'autre: le temps 3. Tov un pep. that he who could not bear misfortune.

ne vous dure guère, car il y a une heure et demie; et je me moque

de ceux qui me disent que le temps me dare à moi, et que j'en 4. Eumpayıa, as, Ý, prosperity, literally well-doing, from er juge par fantaisie : ils ne savent pas que j'en juge par ma montre. and apartev, I do, I am in a certain condition, as in our idem. phrase' “ How do you do ?” περιχαρης, tery joyful, περι gives Beaucoup de gens ne donnent pas leur bien, mais semblent le ihe idea of much or eccess ; opyn, ns, , anger, here used for jeter. Je n'appelle pas libéral un homme qui agit comme s'il adversity, considered as a consequence of the anger of the l'était en colère contre son argent.--Sénéque.

εισι,

τοις

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PLUR. SING.

rejoice.

Persent Tense. Present Tense. Present. 2. freue (du) bich, fich freuen, to fich freuend rejoice thou

rejoice, rejoicing
3. freue (er) fich.
1. freuen (wir)

uns,
2. freuet (ihr)

euch,
3. freuen (sie)

fist

PLUR: SING.

may have

PLUR. SING.

Perfect Tense.
fich gefreut haben,
to have re-
joiced.

gefreut,

gefreut,

Present Tense.

Present Tense. ich freue mich, I rejoice. ich freue mich, I may Bu freuest dich, thou rejoieest. du freuest did, thou mayst er freuet fich, he rejoices. er freuet fich, he may wir freuen uns, we rejoice. wir freuen uns, we may ihr freuet euch, you rejoice. ihr freuet euch, you may fie freuen sich, they rejoice. fic freuen sid, they may Imperfect Tense.

Imperfect Tense. ich freuete mich, I rejoiced. ich freuete midy, I might repu freueteft dich, thou didst rej, du freuetest dich, joice, &c. er freuete fich, he rejoiced. ci freuete fict), wir freueten uns, we rejoiced. wir freueten uns, ihr freuetet cudy, you rejoiced. ihr freuetet euch, fie freueten sich, they rejoiced. fie freucten fich, Perfect Tense.

Perfect Tense. ich habe mich I have re- ich habe mich |du haft bis joiced, &c. du babest dich Irejoiced, er hat sich

er habe sich

&c. wir haben uns

wir haben und iýr habet euch

ihr habet euch fie haben fich

fie haben fich Pluperfect Tense.

Pluperfect Tense. ich hatte mid I had re- id) hätte mid) I might du hatteft dich joiced, &c. du hättest rich

have reer hatte sich

er hätte fic) joiced, &c. wir hatten und

wir hätten ing 2 ihr hattet euch

ihr hättet eude fie hatten sich

sie hätten sich First Future Tense.

First Future Teuse. First Future. ich werde mich I shall re- ich werde mich (if) I shall ich würde mich du wirst dich joice, &c. du werdest dich rejoice, &c. pou würdest dich er wird sich

er werte fie

Jer würde fide wir werden und wir werden und

wir würden uns 2 ihr werdet such

ihr wertet euch

ihr würdet euch 3 fic werden sich

fie werten sich

lfie würden sich Second Future Tense. Second Future Tense. Second Future. 1 ich werde mich I shall ich werde mich (if) I shall ich würde mich 2 tu wirft dich have re- du werbest bich have re-fou würdest dich 3 er wird sich joiced, &c. er werte sich joiced, &c. fer wütbe fich 1 wir wer benung wir werden und

wir würden uns 2 ihr werdet euch

ihr werdet euch

Jihr würdet euch lfie werden fich fte werden fich

lite würden fich

PLUR. SING.

gefreut,

gefreut,

PLUR, SING.

freuen,

freuen,

freuen,
I should re-
joice, &c.

PLUR. SING.

gefreut haben,

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§ 88. IMPERSONAL Verbs,

EXAMPLES (1) The impersonal verb, properly so called, is one destitute

Es ärgert mich, it rexes me, i.e. I am vexed ; of the first and second persons: being confined to the third

c# friert ihn, it chills him, i.e. he is chilled or frozen ;

es hungert mich, it hungers me, i.e. I am hungry; person singular, and having for its grammatical subject the pronoun es, without definite reference to any antecedent, as,

es reift, there is a hoar frost;

es heißt, it is said ; ed regnet, it rains ; es blißt, it lightens ;

68 wird viel davon geretet, it is much talked about; es schneit, it snows; friert, it freezes;

ef verstehet fich, it understands itself, i.e. it is understood; &o: e$ donnert, it thunders ; e8 thaut, it thaws;

48 fragt sich, it asks itself, i.e. it is asked, it is the question ; ed hagelt, it hails; es tagt, it dawns.

£8 giebt Menschen, it gives or yields men, i.e. there are men, (2) It must immediately appear, that a verb may be imper

$ 89. COMPOUND VERBS. sonol, and yet belong to any of the classes of verbs described (1) Various derivative verbs in German are produced by in preceding sections. Thus some are transitive : some are the union of simple words with prefixes. Under the name of intransitive ; some are passive ; some are reflexive ; &c. Prefixes are here comprehended all those invariable words, las

adverbs and prepositions), which are combined with other to which they lead are only more or less approximate in accor words to vary or modify their signification. They are, also, dance with the results of experiment. often called Particles. The simple words with which they are There are sereral cases to be considered in the motion of united, are generally verbs; but often nouns and adjectives are, liquids : viz., the efflux of a liquid-Ist, from an orifice in the by prefixes, converted into verbs. Most of the prefixes are thin wall of a reservoir, the thickness of which is less than separable, that is, may stand apart from the radicals; some, half the smallest dimension of the orifice ; 2nd, from an ori. however, are found to be inseparable ; some are either separable fice of the same kind furnished with an adjutage; 3rd, through or inseparable, according to circumstances.

tubes of large diameter ; 4th, through capillary tubes; and (2) The prefixes are themselves, also, either simple or com- 5th, over a channel, as the beds of rivers. "We shall particupound; as, Herkommen, to come here or hither ; herüberfommen, larly consider four of these cases. to come over here, or hither. In most instances, the prefixes may be translated severally as above; but often they are found Let us first consider the flow of water from the orifice of a

1. Eflux through orifices in a thin wall; and Liquid Vein.to be merely intensive or euphonic. This is likewise often vessel having thin walls or sides. If at any point in such a the case in English: thus, ex (which literally signifies wall we make a small opening, the liquid will issue from it out or out of,) has, in some words the signification very, ex- under the action of two forces : 1st, gravity, which acts upon ceedingly or the like; as, exasperate, to make very angry; so a, it in the vertical direction; and 2nd, the pressure of the liquid, in the word ameliorate is merely euphonic, the derivative form which acts perpendicularly to the wall, and proportionally to (ameliorate) meaning nothing more than the simple one, the depth of the orifice. meliorate.

The jet of the liquid which thus issues from the reservoir is $ 90. SIMPLE PREFIXES SEPARABLE.

denominated the vein. If the orifice is made in the bottom of Ab, from, off, down; Abseßen, to set or put down; the reservoir, the action of gravity being in the same direction

to depose.

as the interior pressure of the liquid, these two forces are An, to, at, in, on, towards; Anfangen, to catch at, i.e. to added together, and the vein is vertical and rectilinear. But

begin.

if the orifice is made in a wall vertical or inclined, the two Auf, on, upon, up;

Aufgehen, to go up; to rise. forces which act upon the liquid are such that the one is Au8, out, out of, from; Ausnehmen, to take out; to vertical, and the other horizontal or oblique in its direction. choose.

In this case, the liquid vein following the direction of the Bri, by, near, with; Beistehen, to stand by; to as- resultant, takes a curvilinear form, which, abstracting the

sist.

resistance of the air, would be exactly that of the curve which there, at ;

Dableiben, to remain there, or projectiles describe in a vacuum, and known under the name at, to stay; to persist.

of the parabola. Dar, there, at;

Darreichen, to reach there, i.e. Structure of the Liquid Vein.- To the investigations of M. to offer.

Savart we owe the following particulars relating to the nature Gin, in, into;

Ginkaufen, to buy in ; to pur- of the liquid vein. It is composed of two distinct parts: the · chase.

first, which is in contact with the orifice, is completely calm Empor up, upward, on high; Emporheben, to lift up. and transparent, and presents the appearance of the most Fort, onward, away, forward; Fortfahren, to drive or bear limped crystal cylinder; the second, on the contrary, is troubled

on; to continue.

and agitated, and presents elongated swells, which are reguGegen, towards, against; Gegenhalten, to hold against; larly arranged at intervals, as shown in fig. 41, and which may to resist ; to compare.

be termed protuberances, In, in, within ; Inwohnen, to dwell in.

This second part of the vein is not Heim home, at home; Heimkehren, to turn home

continuous; for when opaque wards; to return.

liquid, such as mercury, is made to Her, hither, here; Herbringen, to bring hither,

flow through the orifice, we or along.

through the vein. Savart has observed Hin, thither, there, away; Hingehen, to go thither, or

that the protuberances are formed of away

discontinuous globules, elongated in Mit, Mitnehmen, to take with, or

a direction transverse to that of the along.

vein ; and that the contractions or Nach, Nachfolgen, to follow after; to

nodes are formed, on the contrary, of succeed.

globules elongated in the direction of Nieber down, downwards, under; Niederreißen, to pull down.

the vein itself, as shown in fig. 42. Db, on, over, on account of; Obliegen, to lie on, i.e. to ap

He has also observed, by looking at ply one's self to; to be in

the vein in a strong light, that the cumbent on.

limped part is formed of annular for, before, Vorgehen, to go before; to

swells which originate near the ori. surpass.

fice, and are propagated at equal Wegbleiben, to stay away.

intervals until they reach the troubled Bu, to, towards; Zugeben, to give to; to grant.

part of the vein where they are separated. These swells proceed from periodic pulsations which take place

near the orifice. Their number is in ON PHYSICS OR NATURAL PHILOSOPHY.

the direct ratio of the velocity of

efflux, and in the inverse ratio of the No. XIII.

diameter of the orifice, HYDRODYNAMICS.

The pulsations just mentioned may Object of the Science. It has been already stated that hydro

be so rapid as to give rise to a sound, dynamics is that part of Rational Mechanics which treats of the

which is increased by receiving the motion of liquids; and that the part of this science which

vein on any tightened membrane. By particularly treats of the art of conducting and raising water,

producing a sound in unison with that is called hydraulics ; that is, hydraulics is the practical depart

of the vein, by means of a musical ment of hydrodynamics.

instrument, Sayart has modified the In hydrodynamics as well as in hydrostatics, liquids are

Figs. 41,

42. vein in such a manner, that the proconsidered to be incompressible, perfectly fluid, and conse- tuberances and nodes have taken a more regular form, and the quently free from all viscosity. But liquids possess these pro- transparent part of the vein has entirely disappeared. He has perties only imperfectly; hence the theoretical consequences also found that the resistance of the air hes no effect on the

an

see

with;

after ;

Wor,

Weg,

away, off ;

form and dimensions of the vein, or on the number of pulsa- the velocity due to the fall mn, which is the space through tions. He has likewise observed that the structure of the which it would ascend, but for the retarding circumstances just horizontal or oblique veins does not essentially differ from that mentioned. of veins which fall vertically. Vena Contracta, or the Contraction of the Dein. When efflux

Fig. 44. takes place through a circular orifice made in a thin wall or side of a vessel full of water, the liquid vein preserves the circular form in its transverse sections, but the diameter is variable. This diameter is at first equal to that of the orifice, it then rapidly diminishes, and at a distance from the orifice nearly equal to its diameter, the section of the vein is no more than f of that of the orifice. If the direction of the vein is vertical as in fig. 41, the section decreases slowly till it reaches the troubled part. If the direction of the vein is horizontal, the section decreases insensibly. If the angle of inclination of the vein varies from 25° to 450, the vein preserves nearly the same diameter ; but if it exceeds 45°, the section increases from the part contracted to the part troubled. The part where the diameter of the section reaches its minimum, is called the contracted section.

The contraction of the vein originates in the converging directions which the liquid particles assume in the interior of The following important corollaries are deduced from the the vessel, when they proceed towards the orifice. This phe- Theorem of Torricelli : Jst. All bodies in a vacuum falling nomenon is rendered visible by putting the water in a trans- with equal velocity, it follows that the velocity of discharge is parent vessel, and mixing small light substances with it independent of the density of the liquid. For example, water which are kept in suspension in it, the orifice being made in a and mercury issue with the same velocity, if the height of the thin wall or side of the vessel. If the orifice be half an inch level above the orifice be the same for both liquids. Experiin diameter, we see at twice or thrice that distance from it ment, indeed, proves that in the case of equal heights and within the vessel, the substances suspended in the water orifices of the same diameter, equal volumes of these liquids drawn from all parts of the vessel towards the orifice, and are discharged in the same time. 2nd. The velocity of disdescribing curve lines, as if they were attracted towards a centre, charge at the issue of a liquid from the orifice, is proportional as shown in fig. 43. The convergence of the parti which to the square root of the height of the level in the reservoir

above the centre of the orifice. This is, in fact, a consequence Fig. 43.

of the laws of gravity, for we have seen, in a former lesson, that representing the velocity, acquired by a moveable which falls in a vacuum by v, and the height of the fall by h, we have v=V/2gh. The velocity calculated by this formula is called the theoretical velocity.

Theoretical and Effective Discharges. - The voluine of a liquid which is actually discharged from an orifice in one second, is called its effective discharge ; and the volume of a liquid equal to that of a cylinder or prism which has the orifice for its base, and the theoretical velocity above mentioned for its height, is called the theoretical discharge.

The effective discharge is always less than the theoretical

discharge. The effective discharge is in reality the product took place in the interior of the vessel is continued exteriorly, of the contracted section of the vein, and the mean velocity of and the liquid vein is gradually contracter till it reaches the the liquid particles at the instant that they pass this section. point where the particles, by the effect of their mutual action, take a parallel or diverging direction. The vein thus forms a

Fig. 45. species of truncated cone or frustrum, of which the greater base is the orifice, and the smaller base the contracted section.

In the preceding remarks we have supposed that the orifice is of the circular form. If it be polygonal, or of any form different from that of a circle, the vein no longer preserves a section of the same form as the orifice. Its form changes as the vein recedes from the orifice, and continually gives rise to protuberances and nodes.

Theorem of Torricelli.—When a liquid issues from a reservoir by an orifice in a thin wall or plate, the velocity of the discharge is determined by the following theorem : The liquid particles as they issue from the orifice have the same velocity as if they fell freely in a vacuum, from a height equal to the vertical distance from the centre to the upper surface of the liquid in the re-ervoir. This theorem was discovered by Torricelli in 1643, and was by him considered as a corollary to the laws of falling bodies established by his master, Galileo. This law can be experimentally proved to be a result of the principle demonstrated in mechanics, viz., that when a body is projected upwards with a given velocity, it will rise to the same height from which it would have fallen in order to acquire that velocity. Thus, when the discharge is made to take place vertically upwards, as represented in fig. 44, the liquid vein reaches very nearly the height of the level of the liquid in the vessel from which it is discharged, and the reason why it does not reach it If the area of the section were the same as that of the orifice, entirely, is the resistance of the air, and the action of the liquid and if the mean velocity were the same as the theoretical particles in falling, which oppose the ascent of the jet. Hence, velocity, the effective discharge would be the same as the at its issue from the orifice n, the liquid spouts upwards with theoretical discharge; but it generally happens either thas the

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