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LESSONS IN FRENCH.-XXXI.

SECTION LVII.-IDIOMATIC PHRASES.

1. CHANGER [1, see § 49 (1)], used in the sense of to change, to
leave one thing for another, is followed by the preposition de:
changer d'habit, de chapeau, etc., to put on another coat, hat,
etc.; changer d'avis, to change one's mind; changer de maison,
to move, to change houses; changer de place, changer de pays,
changer de climat, to go to another place, country, climate;
changer de nom, to change one's name. The student will per-
ceive that the noun following changer is not preceded by a
possessive adjective, like the noun of the English sentence.
Voulez-vous changer d'habit?
Ce monsieur a changé de nom,

2. Changer contre means to change for, to get change for.

Will you change your coat?

That gentleman has changed his name. exchange for; changer pour, to

France? 12. I long to be there. 13. Does not your mother
tarry too long? 14. She is very long in coming. 15. Have
you changed the forty-franc piece? 16. I have not changed it
yet. 17. Why have you not changed it? 18. Because your
father has no change. 19. Have you the change for a guinea?
20. No, Sir, I have only twelve shillings.

SECTION LVIII.-RULES FOR THE PLURAL OF COMPOUND
NOUNS.

1. We have given, in Section IX., rules for forming the plural of nouns, but, in accordance with our plan of not presenting too many difficulties at once, we have deferred until the present section the rules for the formation of the plural of compound

nouns.

2. When a noun is composed of two substantives, or of a substantive and an adjective, both take the form of the plural: un chef-lieu, des chefs-lieux, a chief place, chief places; un

Voulez-vous changer votre chapeau Will you exchange your hat for gentilhomme, des gentilshommes, a nobleman, noblemen (§9 coutre le mien ?

mine?

Changez ce billet pour de l'argent, Change that note for silver.

3. Tarder means to tarry, to be long in coming. Tarder used unipersonally, and preceded by an indirect object, means to long, to wish for.

Votre sœur tarde bien à venir,
Il me tarde de la voir,

RÉSUMÉ OF

N'avez-vous pas changé d'apparte.

ment?

Nous avons changé de maisons.
Votre frère a changé de conduite.

Your sister is very long coming.
I long to see her.

EXAMPLES.

(1) (3)].

3. When, however, two nouns are connected by a preposition, the first only becomes plural: un chef-d'œuvre, des chefsd'œuvre, a master-piece, master-pieces [§ 9 (2)].

4. In words composed of a noun and a verb, preposition or adverb, the noun only becomes plural: passe-port, passe-ports, passport, passports [§ 9 (6)].

5. Words composed of two verbs, or of a verb, and adverb, and a preposition, are invariable: un passe-partout, des passeHave you not taken another apart partout, master-key, master-keys [§ 9 (8)].

ment?

duct.

We have changed houses. Your brother has changed his conContre quoi avez-vous changé votre For what have you exchanged your cheval ? horse?

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6. We have seen [Sect. III. 4] that the name of the material always follows the name of the object, and that both are united by the preposition de. The name of the profession or occupa tion also follows the noun representing the individual, and the same preposition de connects the two: un maître d'armes, a fencing-master; un maître de dessin, a drawing-master; un marchand de farine, a dealer in flour [§ 76 (12), § 81 (4)].

7. The name of a vehicle, boat, mill, etc., always precedes the noun describing the power by which it is impelled, or the purpose to which it is adapted; the name of an apartment, that of

Il leur tardait de revoir leurs amis. They longed to see their friends the use to which it is appropriated. The connecting preposition

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is à un moulin à vapeur, a steam mill; un bateau à vapeur, a
steamboat; un moulin à eau, a water-mill; la salle à manger,
the dining-room [§ 76 (13), § 81 (5)].

RÉSUMÉ OF
Lille et Arras sont les chefs-lieux
des départements du Nord et du
Pas-de-Calais.

Les chemins de fer et les bateaux
à vapeur sont très-nombreux en
Amérique.

Cette maison contient un salon,
une salle à manger, une cuisine,
et plusieurs chambres à coucher.

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1. Cet homme n'a-t-il pas changé de vie? 2. Il a changé Les moulins à vent sont plus com- Windmills are de conduite. 3. Cette grande maison n'a-t-elle pas changé de maître? 4. Elle a changé de maître, le Capitaine G. vient de l'acheter. 5. Vous êtes mouillé, pourquoi ne changez-vous pas de manteau ? 6. Parceque je n'en ai pas d'autre. 7. Votre cousine ne change-t-elle pas souvent d'avis? 8. Elle en change Armes, f. pl., fencing. | Dessin, m., drawing. bien souvent. 9. Pendant le combat, ce jeune soldat n'a-t-il pas changé de visage? 10. Il n'a point changé de visage. 11. Ce malade ne devrait-il pas changer d'air ? 12. Le médecin lui recommande de changer de pays. 13. Où est votre cheval gris ? 14. Je ne l'ai plus, je l'ai changé contre un blanc. Avec qui l'avez-vous changé ? 16. Je l'ai changé avec le jeune homme qui demeurait ici le mois passé. 17. Le marchand peut-il me changer cette pièce de quarante francs? 18. Il ne saurait (cannot) vous la changer, il n'a pas de monnaie. 19. Avez-vous la monnaie d'une guinée (change for a guinea) ? EXERCISE 110.

15.

1. Why do you not change your coat? 2. For a very good reason (raison, f.), because I have no other. 3. Has your father changed houses? 4. No, Sir, but we intend to do so (de le faire) to-morrow. 5. Has that child altered his conduct? 6. He has altered his conduct, he is very good now (maintenant). 7. Was er afraid; did not his countenance change? 8. changed, but he was not afraid. 9. Have you (chambre, f.)? 10. I have not changed ory good. 11. Do you not long to be in

not xon

T

1. Faut-il avoir un

Se munir, 2 ref., to pro-
vide one's self with.
EXERCISE 111.

passe-port pour voyager en France? 2 Il faut en avoir un. 3. Les Anglais se munissent-ils de passeports pour voyager en Angleterre ? 4. On n'a pas besoin de passe-port en Angleterre. 5. Aimez-vous à voyager sur les chemins de fer? 6. J'aime mieux voyager sur les chemins de fer que sur les chemins ordinaires. 7. Avez-vous apporté vos passe-partout? 8. Je n'ai point de passe-partout, je n'ai que des clefs ordinaires. 9. Votre frère est-il venu dans un bateau vapeur? 10. Il est venu dans un bateau à voiles. 11. Aver vous une voiture à quatre chevaux? 12. Non, Monsieur, nous n'avons qu'un cabriolet à un cheval. 13. Votre frère a-t-il bati un moulin à vapeur? 14. Il a fait bâtir deux moulins, l'anà vent et l'autre à eau ? 15. Votre compagnon a-t-il engage in maître d'armes? 16. Non, Monsieur, il a déjà un maître de dessin et un maître de danse. 17. Combien de chambres à

coucher avez-vous ?

18. Nous en avons deux. 19. Avez-vous 27. Cette belle petite fille n'a ni faim ni soif. 28. Qu'a-t-elle ? 29. une bouteille de vin ? 20. Non, Monsieur, mais j'ai une bouteille Elle n'a ni parents ni amis. 30. Cette montre d'or est-elle boune ? 32. L'avez-vous ? à vin (wine-bottle) [§ 81]. 21. Voyez-vous les chats-huants? 31. Celle-ci est bonne, mais celle-là est meilleure. 33. Je l'ai, mais je n'ai pas celle de votre sœur. 34. Je n'ai ni la vôtre 22. Non, mais je vois les chauves-souris. ni la mienne, j'ai celle de votre mère.

EXERCISE 112.

1. Is your father in England? 2. No, Sir, he is in France

3.

with my brother. they have taken two. 5. Is it necessary to have a passport to travel in America ? 6. No, Sir, but it is necessary to have one to travel in Italy. 7. Is there a steamboat from Calais to Dover (Douvres)? 8. There are several. 9. Is there a railroad from Paris to Brussels (Bruxelles)? 10. There is one from Paris to Brussels, and one from Paris to Tours. 11. Has your brother bought a windmill ? 12. No, Sir, but he has built a steam-mill. 13. Are there many wind-mills in America? 14. No, Sir, but there are many water and steam-mills. 15. Does your cousin learn drawing? 16. He does not learn it, he cannot find a drawing-master. 17. Is the fencing-master in the aining-room? 18. No, Sir, he is in the drawing-room. 19. Is your cousin in his bedroom? 20. No, Sir, he is out (sorti). 21. How many rooms are there in your house? 22. Five: a kitchen, a dining-room, a drawing-room, and two bedrooms. 23. Are there owls here? 24. Yes, Sir, and bats too.

Have they taken passports? 4. Yes, Sir,

KEY TO EXERCISES IN LESSONS IN FRENCH.
EXERCISE 18 (Vol. I., page 78).

1. Avez-vous mes tables ou les vôtres ? 2. Je n'ai ni les vôtres ni les miennes, j'ai celles de l'aubergiste. 3. Les avez-vous ? 4. Non, Monsieur, je ne les ai pas. 5. Votre sœur a-t-elle mes chevaux ? 6. Oui, Monsieur, elle a vos deux chevaux et ceux de votre frère. 7. Avez-vous raison ou tort? 8. J'ai raison, je n'ai pas tort. 9. Le ferblantier a-t-il mes chandeliers d'argent ou les vôtres ? 10. Il n'a ni vos chandeliers d'argent ni les miens. 11. Qu'a-t-il? 12. Il a les tables de bois de l'ébéniste. 13. A-t-il vos chaises d'acajou? 14. Non, Monsieur, il a mes tables de marbre blanc. 15. Avez-vous ces tablesci ou celles-là? 16. Je n'ai ni celles-ci ni celles-là, j'ai celles de l'ébéniste. 17. Avez-vous de bons porte-crayons ? 18. Non, Monsieur, mais j'ai de bons crayons. 19. Le voyageur a-t-il des fusils de fer? 20. Oui, Monsieur, il a les miens, les vôtres et les siens. 21. N'a-t-il pas ceux de votre frère? 22. Il n'a pas ceux de mon frère. 23. L'ouvrier a-t-il mes marteaux de fer? 24. Oui, Monsieur, il les a. 25. Mon frère a-t-il vos plumes ou celles de mon cousin ? 26. Il a les miennes et les vôtres. 27. Avez-vous les habits des enfants? 28. Oui, Madame, je les ai. 29. Avez-vous le chapeau de votre sœur? ma cousine. 31. Votre frère a-t-il quelque chose? 32. Il a froid et faim. 33. Avez-vous des chevaux ? 31. Oui, Monsieur, j'ai deux chevaux. 35. J'ai deux matelas de crin et un matelas de laine.

EXERCISE 19 (Vol. I., page 79).

30. J'ai celui de

1. Is that lady pleased? 2. No, Sir, that lady is not pleased. 3. Is your daughter quick? 4. My son is very quick, and my daughter is idle. 5. Is she not wrong? 6. She is not right. 7. Is your cousin happy? 8. Yes, Madam, she is good, beautiful, and happy. 9. Has she friends? 10. Yes, Sir, she has relations and friends. 11. Has she

a new dress and old shoes? 12. She has old shoes and an old dress. 13. Has not your brother a handsome coat? 14. He has a handsome coat and a good cravat. 15. Have you good meat, Sir? 16. I have excellent meat. 17. Is this meat better than that? 18. This is better than that. 19. Has your friend the beautiful china inkstand? 20. His inkstand is beautiful, but it is not china. 21. Is any one hungry? 22. No one is hungry. 23. Are the generals here? 24. The generals and the blacksmiths are here. 25. I have your parasols and your umbrellas, and your children's.

contente.

EXERCISE 20 (Vol. I., page 79).

1. Votre petite sœur est-elle contente ? 2. Oui, Madame, elle est 3. Cette petite fille est-elle belle ? 4. Cette petite fille n'est pas belle, mais elle est bonne. 5. Avez-vous de bon drap et de

bonne soie? 6. Mon drap et ma soie sont ici. 7. Votre sœur estelle heureuse? 8. Ma sœur est bonne et heureuse. 9. La sœur de ce médecin a-t-elle des amis ? 10. Non, Madame, elle n'a pas d'amis. 11. Votre viande est-elle bonne ? 12. Ma viande est bonne, mais mon fromage est meilleur. 13. Le libraire a-t-il un bel encrier de porcelaine? 14. Il a un bel encrier d'argent et une paire de souliers de cuir. 15. Avez-vous mes parasols de soie? 16. J'ai vos parapluies de coton. 17. L'habit de votre frère est-il beau ? 18. Mon frère a un bel habit et une vieille cravate de soie. 19. Avez-vous des parents et des amis ? 20. Je n'ai pas de parents, mais j'ai des amis. 21. Cette belle dame

a-t-elle tort ? 22. Cette belle dame n'a pas tort. 23. Avez-vous de belle porcelaine? 24. Notre porcelaine est belle et bonne. 25. Elle est meilleure que la vôtre. 26. Cette petite fille n'a-t-elle pas faim ?

MECHANICS.-XIV.

ILLUSTRATIONS OF PRECEDING PRINCIPLES-KITE, BOAT, ETC. -ELEMENTS OF MACHINERY.

WE have now to trace the practical application of the principles already laid down, and the best way of doing this is to take some common instances and carefully examine them, and we shall see that the same rules will apply to other and more complicated cases.

A heavy box is resting on a four-legged table. What are the forces that act on it, and what on the table? On the box there are only two-its own weight acting downwards, and the upward pressure of the table, exactly counterbalancing this weight. We turn, then, to the table, the forces acting on which are not quite so casily determined. There are its own weight and the weight of the box acting through their respective centres of gravity. These are parallel forces, and, as we have seen, have a resultant equal to their sum, and acting at a point in the line joining them, so taken that their distances from it are in the inverse proportion to their intensities. The other force which acts on the table is the resistance of the floor on which it rests, which resistance is transmitted upward through the four legs. If the weight act at a point equally distant from these, each bears an equal share; but if not, it is divided between them in the inverse proportion to their distances. To make this more clear, we will suppose these distances to be 2, 6, 6, and 8 feet respectively. Find the least common multiple of these numbers, that is, the least number each will divide without any remainder. In this case it is 24. This we divide successively by the distances, and obtain the quotients 12, 4, 4, and 3; and these numbers represent the proportion in which the weight is divided between the legs.

Now suppose the weight of the table and box to be 207 pounds. Since 12, 4, 4, and 3, added together make 23, the leg 2 feet off supports 12 parts out of every 23, i.e., of the weight, or 108 pounds. Those 6 feet off support, or 36 pounds each; and the other sustains only, or 27 pounds. A calculation of this sort is mine the relative strength the difvery frequently required to deter R ferent parts of a building should have.

A

W

В

Fig. 86.

We will take another case. body, a (Fig. 86), rests on an inclined plane, the angle of which, A. C A B, is 30°, and the co-efficient of friction is . What forces act on G, what are their amount, and what other force must be applied to keep it in its place? We will try and solve these questions. As already seen, three forces act on G-its weight acting along G w, the resistance of the plane acting along G R, and the force of friction, which is of the weight and acts along F G. We found, when considering the inclined plane, that the power necessary to sustain o must bear the same proportion to its weight that B C does to A C. This, then, is the first thing we must find out, and we must have a slight acquaintance with mathematics for this; there is, however, no difficulty in the matter. Produce C B to D, so as to make B D the same length as B C, and join A D. The triangles ABC and ABD are exactly equal. For as B C is equal to B D, and each is at right angles with AB, A B C would, if we were to turn it over, exactly lie on A B D. AD is, then, equal to a C. Now in any and every triangle the three interior angles are together equal to 180°, or two right angles, and in ABC we know that the angle CA B is 30°, and ABC, being a right angle, is 90°; therefore ACB must be 60°, and A D B is equal to it, and therefore is also 60°. The angle B A D is likewise equal to BA C, and as each is 30°, the angle CAD is 60°. We see thus that each of the angles of the triangle CAD is 60°, and therefore they are equal to one another; and, since the angles are equal, the sides are also equal, for there is no reason why one should be greater than another. The triangle is thus equilateral and equiangular. We have now found out what we wanted; for, if вo be repre

Here is another case, involving the same principle. A bracket

scated in length by 1, CD will be 2; and AC is equal to CD, therefore it is also 2, and the proportion BC bears to A c is 1A B (Fig. 88), projects from a wall, to which it is fastened by a

2. or On an incline of 30°, then, the power must be half the weight; but, in this case, friction sustains one-fourth, and therefore a power must be applied, acting in the direction GP, and equal to one-fourth of the weight, in order to maintain equilibrium.

We have now to solve the remainder of the question. We know how much GW, GF, and GP are; but we want to know what portion of the weight is borne by the plane, that is, what proportion AB, which represents the resistance of the plane, bears to AC. To find this, we need another very important geometrical proposition, which you will find fully proved in Lessons in Geometry, Problem XXX., Vol. I., page 337.

In every right-angled triangle the square described on the side opposite the right angle is equal to the sum of the squares on the sides containing it. If we represent the sides by numbers expressing their lengths, the rule holds equally true. Suppose, for example, we measure off from one of the sides containing the right angle a length of 4 inches, and from the other a length of 3 inches, we shall find the line joining these two points is equal to 5 inches.

The square of 4 (which means 4 multiplied by itself) is 16, and the square of 3 is 9. These added together make 25, which is the square of 5. The usual way of writing this is 42+32= 52. In this way, if we know the length of any two sides of a rightangled triangle, we can always calculate the third. Now in the case we are examining, we know that the side a C is equal to 2 and the side B C to 1; but the square of AC is equal to the sum of the squares A B and B C; the square of A B must, therefore, be equal to the difference between those of A C and BC. Now these are 4 and 1; the square of A B is, then, equal to 3, and the length of A B must be represented by the quantity which, multiplied by itself, will make 3. This is called the square root of 3, and is written 3. By arithmetic we can easily find exactly what this number is, but you can see that it is very nearly 13. The proportion, then, of A B to A c is 12 to 2, or 7 to 8, and the plane sustains a pressure equal to about of the weight. We have thus discovered the magnitude of all the forces as required. When, as in our last lesson, we have resolved all the forces acting on a body along two lines at right angles, we can in this way find the magnitude of the resultant without the trouble and possible inaccuracy of actual measurement. Suppose we have a remainder of 12 pounds acting along one of the lines, and one of 5 pounds along the other, the resultant will be equal to 12+5; that is, to the square root of 144 + 25, or 169, which is 13. In the same way we can solve many questions frequently met with. Here is an example. Two forces act on a body; the resultant is 34 pounds, and one of the forces is 16 pounds; what is the other? We first find the square of 34, which is 1,156; from this we take the square of 16, or 256, and we have left 900. The square root of this is 30, and this accordingly is the intensity of the other force.

Now turn to another common thing. A ladder, A B (Fig. 87), leans against a wall. What are the forces acting on it? Its own weight acts vertically downwards through G, and the other forces which keep it at rest are the reaction of the wall and the ground! at A and B respectively. Now there is but little friction at A, and we may therefore consider the reaction to be in the

W

screw. A strut, A C, supports the outer end, and a weight, w,
rests on it. In what direction is the strain on the screw? The
three forces here are, gravity acting along w G, the thrust of
the beam, A C, and the strain on the screw. The two former
act through the point o, and the direc-
tion of the third is found by drawing a
line from this point to the screw. We
may take a line, o c, of such a length
as to represent the weight, and resolve
it into o b and o a, one acting along
the strut, the other perpendicular to
the wall. These will represent two
forces, which are together equivalent to
o c, and of these o b will be overcome
by the pressure of the strut, and the
other force, o a, tends to draw the
screw from the wall. A portion, how-
ever, of the pressure of the strut will
be borne by the screw, and these two
forces combining produce the resultant,
which acts on the screw towards the
point o.

a

Fig. 88. It is frequently very important to be able thus to tell in what direction a strain will act, as the strength of our materials must be proportionate to it. In this way the direction of the tie-beams and king and queen posts of a roof are determined. We know, too, where to apply struts and braces to the framework of a building, so as to gain the greatest benefit from them.

Now there are one or two cases of the composition and resolution of forces that are frequently given as illustrations, and if we clearly understand them we shall be able to master most others. The first is that of a kite, for there is science to be learned even from this common plaything. Indeed, we shall frequently find, among the very commonest things, good illustra tions of any subject we may be studying.

Let K (Fig. 89) represent a kite. The forces which act on it are, the force of the wind, acting, we will suppose, in the direction of the arrow, the tension of the string, acting along it in the direc tion K s, and the weight of the kite; and by the action of these three forces it is kept at rest. We will consider them singly and first, we will take the force of the wind. Take K w, of such a length as to represent this force. The kite is always so made as to present a large surface to the wind in proportion to its weight, and the string is fastened to the loop in such a way that it does not hang vertical, but inclined at an angle; the tail, however, prevents its being so acted on by the wind as to come

Fig. 89.

direction AP perpendicular to the wall. GW and AP, then, represent two of the forces acting on the ladder; but, as it is at rest, all three forces must act through the same point. Now the only point in which wG and AP meet is that found by producing w G till it cuts A P. Let R be this point; the force in the same straight line with the cord. Let us, then, resolve at в must act in the direction B R. This force the force of the wind into two, one acting edgewise on the kite, is the resultant of two others the resistance the other perpendicular to its surface. We draw the paralleloof the ground acting vertically upwards, and gram KOWA, and thus have the two forces, K O and K A, instead the force of friction which acts along the of K W. Ko has no effect, as it acts on the edge, and we need, ground and towards the base of the wall; and, therefore, only consider the part K A. Fig. 87. we easily see that the more nearly vertical the ladder is, the greater is the former as compared with the latter, and therefore the less the amount of friction which is required to keep it in its place. We have thus the scientific reason for the well-known fact that the more a laria inelined, the greater need there is for the foot to be as to keep it from slipping.

Now we will introduce a second force, that of the string. Produce s K backward, draw A B perpendicular to B K, and complete the parallelogram DK B A. We can again resolve KA into K D and K B. The latter will be expended in stretching the string, and have no tendency to move the kite, and thus we have K D left as the effective resultant of these two forces. We now consider the third, which is the weight of the kite. Draw & @

S

of such a length as to represent this, and complete the parallelogram D C G K. We have then K c the resultant of K D and K G, and therefore of all the forces which act on the kite, and this is the direction in which the kite will move, but as it does so, the angle at which it is inclined varies till K D and K G become opposite and equal, and then the kite will remain at rest as long as the force of the wind remains unaltered.

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Fig. 90.

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The other case we will consider is that of a ship, which will sail within a few points of the wind. Let co (Fig. 90) represent the direction and intensity of the wind, and s v the direction in which it is desired that the vessel should advance. The sail is placed in the direction A B, which is midB way between that of the wind and that of the vessel. We must, as before, resolve c o into two forces, E O and F 0. The part E O, which acts along the direction of the sail, has no effect in moving the vessel; F 0, which acts perpendicularly to the sail, is the effective portion. We must now again resolve this force along two directions, one being that in which the boat moves, the other at right angles to it. We make a o equal to F O, and about it describe the parallelogram HGIO, and thus have two forces, represented by o H and O I, in the place of the original force, c o. Now, of these, o I has no tendency to cause the vessel to advance; it acts sideways on the vessel, usually inclining it, and causing a slight motion, but it is resisted by the pressure of the water against the side; the other portion, o H, represents the portion of the force of the wind which is effective, and produces motion. In the same way you can calculate what portion of the force of the wind is effective in turning a windmill. The vanes are always set at an inclination with the plane in which they turn, and you must resolve the force along two directions, one perpendicular to the surface, and the other along it. The former you again resolve, and thus find what part of it produces rotation, and what part presses against the face of the mill.

ELEMENTS OF MACHINERY.

As it is our object to make these lessons as practical as possible, it will be well to look at a few of the simpler modes of altering and transmitting power. Sometimes this is advanced to the rank of a separate science, and called kinematics, or the science of motion, but it should be referred to here as a part of practical mechanics.

We seldom have our power available for use in the exact way we desire. Sometimes we have an alternate motion, like that of the piston-rod of an engine, and we want to derive from it a rotatory or progressive motion; or we want to transmit it along a direction making some angle with its course, or to make many other alterations in its mode of action.

In large factories there is frequently a long shaft running along the building, and driven by an engine. From this it is required to drive all the machines in the place. This is accomplished by fixing wheels on the shaft, and letting endless straps pass over these and then over the driving pulleys of the machines. The motion may often be greatly altered in this way. The strap itself merely transmits the power, and whether there is a gain or loss in speed or power depends on the comparative size of the sheaves. Frequently there are several of these wheels of different sizes fixed on the axle and on the machine, and thus the speed may be altered at pleasure. If the strap passes over a large one on the shaft and a small one on the machine, there will be an increase of speed, and if we reverse the condition there will be a loss. A common illustration of a similar arrangement is seen in a watch. The spring when fully wound up exerts a much greater power than it does when the watch has run nearly down. Now this would make it go irregularly, and therefore the fusee is introduced. When the force of the spring is greatest the chain acts on the smallest part of the fusee, and therefore has only a short leverage, but as it unwinds and loses its force the chain acts at a greater leverage, and a uniform rate of motion is thus maintained.

Fig. 91.

Sometimes toothed wheels are used instead of straps, especially when the distance through which the power has to be transmitted is small. The advantage is that they do not slip, as straps are liable to do; the friction with them is, however, greater. If we want to transmit motion from a shaft to another placed at an angle with it (Fig. 91), we employ what are known as bevelled wheels. The action of these will be clear from the figure, without any explanation.

Often it is required to change a rotating motion into a progressive one, and we can accomplish this by means of a rack and pinion. A number of notches are cut in a bar of metal (Fig. 92a), of such a size and at such distances that the teeth of the wheel exactly fit into them, and as the wheel is turned the rack is moved onwards. This is very frequently employed when slow and regular motion is required, as in the adjustment of the tube of a microscope. Instead of a rack a chain is sometimes used (Fig. 92 b), the links being made of a peculiar shape, so that the teeth of the wheel may catch in them.

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Fig. 92.

The crank (Fig. 93) is, perhaps, one of the most common of these elements of machinery. The piston-rod of an engine is usually jointed, and the end of the jointed part fixed to an arm projecting from the axle to be turned, and called a crank. Sometimes it is fixed to a pin in one of the spokes of the wheel, but the action is just the same. The force, however, with which it drives the wheel is continually varying. When the piston is at the bottom of the cylinder the crank and piston-rod are in one straight line, and therefore all the power presses on the bearing of the axle, and is lost. As the wheel turns the power acts with a leverage which increases till the wheel has made nearly onefourth of a revolution; it is then at its maximum, and diminishes till the piston-rod reaches its highest point, when it is all again lost. Now it is clear that unless we have some means of regulating the speed the machine will work very unevenly, and at times stop altogether. To obviate this, a large and heavy wheel, called the fly-wheel, is fixed on the axle of the crank. This, when once started, acquires an amount of momentum or moving force which carries it over the dead points when the power is lost. On account of the weight of the wheel, its motion is but slightly accelerated when the piston acts at its greatest leverage, but the additional force is stored up in it, and thus ensures a steady motion. The heavy wheel of a foot-lathe serves precisely the same purpose. The power is here applied during rather less than one-half of the revolution, but the momentum then acquired carries it through the remaining part.

EXAMPLES.

Fig. 93.

1. Forces of 9 and 12 act at right angles; what is their resultant? 2. The resultant of two forces which act at right angles is 10 pounds. One of the forces is 6; find the other.

3. Two men, one on each side of a stream, tow a barge. The angle the two ropes make is 60°, and each pulls with a force of 100 pounds. What is the total force exerted on the barge?

4. The tension of a wire in a piano is 100 pounds, its length is 5 feet. What force is required to draw its middle point 2 inches out of its position?

5. A weight of 90 pounds rests on a plane inclined at an angle of 30°. The co-efficient of friction is }. What force is required to keep it at rest?

ANSWERS TO EXAMPLES IN LESSON XIII.

1. The forces acting at the longer end are the power of 10 pounds acting at a distance from the fulcrum of 6 feet, and the weight of the lever, which is also 10 pounds, and acts through its middle point, or The moments on this end are thus 10x6, 2 feet from the fulcrum. or 671, and 10x 21 or 27. These make 95 pounds. As w acts at a distance of 14 feet, it must be or 76 pounds. 11'

95

2. Since is lost by friction, we may regard the weight as 12 pounds only. Now of the weight of the first pulley is supported by the power, of the next, and and of the other two; and since each

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LESSONS IN

ARITHMETIC.-XXVIII.

FRACTIONS IN CONNECTION WITH COMPOUND
QUANTITIES.

1. To find the Value of any Fraction of a Compound Quantity. It is evident that we have only to divide the given compound quantity by the denominator, and then multiply by the numerator. The first part of the process determines the magnitude of the equal parts into which the denominator indicates that the quantity is to be divided, and the latter takes as many of those parts as are indicated by the numerator.

Thus, of £1 = 3 x 20s. = 15s. Again, of £2 is of 40s. = 4 × 198. = 4 × 8 s. 103s.

25 minutes =

4 hours

of an hour of a day

=

25 60 × 24 of a day.

of a day.

25

days,

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of a day.

Therefore, 3 days 4 hours 25 minutes are 3 + 1 + or 3 of a day, or of a day. Therefore, of 3 days 4 hours 25 minutes = 0 1% of a day.

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-8.

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2.

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3

Again, of a shilling is of 12 pence, or 2 x yd. = 8d. Therefore, of £2 = 10s. 8d.

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of 3s. 2d.

of 1 oz. Troy; of 1 cwt. of 1 yard; of 1 rod.

of 1 mile; of 1 gallon; of 1 peck.

of 1 hour;

of 1 minute; of 1 degree.

7. 3 of of a mile; of of a week.

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Therefore, of 3 days 4 hrs. 25 min. is 1 day 6 hrs. 34 min.

3. To reduce one Compound Quantity to a Fraction of any other.

Finding what fractional part of one compound quantity another given compound quantity is, is called reducing the latter quantity to the fraction of the first.

Thus, finding what fraction of one pound 6s. is, is reducing 6s. to the fraction of a pound.

This is, in fact, only another name for performing the operation of dividing one compound quantity by another, or of finding the ratio of two compound quantities (see Art. 11, Lesson XXVII., page 101).

4. EXAMPLE.-What fraction of £1 7s. 6d. is 3s. 6d. ?

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10. Find the value of 11. Find the sum of

of £5 7s. 3 d. 12. And of of 16s. 6d. 13. Find the value of 14. Find the value of

of 2s. 6d. + of £3 2s. 6d. + }

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15. Find the value of

of

Reduce to the fraction of a pound

16. 438.; 4s. 7d.; 9s. 21d.

17. 13s. 0d.; 34d.; £3 15s. 91d. 18. What part of £1 is of a penny?

19. What part of 1 lb Troy is 7 oz.!

20. Reduce

21. Reduce

of a quart to the fraction of a gallon. of 1 secon to the fraction of a week. 22. Reduce £3 17s. 11 d. to the fraction of £7 35. 23. Reduce 3 pecks 2 gallons to the fraction of 2 bushels. 24. Reduce 15 cwt. 65 lbs. to the fraction of 2 tons 3 cwt. 25. Reduce 1° 15′ 10′′ to the fraction of a right angle. 26. Reduce 1 acre to the fraction of 5 acres 2 r. 40 p. 27. Reduce aty of £1 to the fraction of a penny. 28. Reduce of a week to the fraction of a minute. 29. Reduce of of £2 8s. 9d. to the fraction of £1 1s. 8d. 30. Express as a fraction of £10 the difference betwen £8) and of £8. 31. Find the value of

32. Find the value of

136 gals. 2 qts. 178 gals. 3 qts.

33. Find the value of

1 lb. 7 oz. 4 dwts. 2 lb. 7 oz. 10 dwts.

of 15 guineas.

77 dys. 4 hrs. 30 min. of

6 dys. 12 hrs.

of 517 square feet 72 inches. £3 18s. 8d. £6 12s. 9d.

1° 17 17 15

of

of 104 yards 9 inches.

KEY TO EXERCISES 44, 45, 46, LESSONS XXVI., XXVIL

1. £9 28. 83d.

2. £3 58. 104d.

3. 8 cwt. 1 qr. 6 lbs. 10 oz.

4. 21tons1 cwt. 83lbs. 5. 19 miles 289 rods 2 feet.

6. 1 league 1 m. 7 fur.

10 rods 12 feet.

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