« ForrigeFortsett »
perature, and consequently, whatever may be the density of never falls so low as 100°, even after the greatest rains, the vapour.
During the greatest droughts, it rarely rises about 30°. When Hair hygrometers present many inconveniences in their con- we rise in the atmosphere, with this instrument, the index struction. When constructed of hairs of different kinds, their generally marches towards 0°. In the aerostatic ascent of indications sometimes vary several degrees, although they Gay-Lussac, to the height of 22,966 feet, the hygrometer stood agree in the two fixed points. Besides, the same hygrometer at 269, a number which nearly correeponds to the hygrometric does not always give the same indications; because the hair state ß. Experience has shown that hair hygrometers only permanently lengthens in consequence of the prolonged tension agree with each other, when the hairs employed are of the of the weight which it carries. The best system of graduation, same kind, and are prepäted in the same manner; so that in therefore, is to assume an arbitrary zero point, from which we order to obtain exact indications; à particular table must be can determine, from time to time, the position of the points of constructed for each hygrometer. extreme dryness and extreme humidity; still, even when these Condensation Hygrometers. The object of the condensation conditions are satisfied, the hair hygrometer has the inconve- hygrometers is to show, by the reduction of the temperature in nience of not giving immediately the hygrometric state of the the air, at what point the vapour which it contains would be air. Gay-Lussac constructed a table, however, which shows sufficient to saturate it; such is the principle on which ate the hygrometric state of the air according to the indications founded the hygrometers invented by Mr. Daniell and M. of the hair hygrometer. Experience proves that these indica- Regnault. The hygrometer of Daniell is composed of two tions are not proportional to this state. Thus, when the index glass bulbs connected by a double bent tube, àg. 201. The marks 50o, the number which corresponds to the middle of the scale, the air is far from being half saturated. It was, there
Fig. 201. fore, necessary to find experimentally the hygrometric state corresponding to each degree of the instrument. Gay-Lussac resolved this problem, by referring to the principle, that the vapours furnished by a saline or acid solution have a maximum tension which is weaker in proportion to the greater quantity of salt or acid dissolved, the temperature being equal. He accordingly placed the hair hygrometer under a bell-glass, in which there was put a mixture of water and sulphuric acid, and he observed the degree of the hygrometer when the air of the glass was saturated. In order, then, to obtain the tension of the vapour under the glass, he put into the vacuum of a barometer, some drops of the same saline solution which he had put under the glass. The depression of the mercury in the barometer then gave him the tension of the mercury under the bell-glass; because in the state of saturation, and at the same temperature, the elastic force of a vapour is the same in a vacuum as in the air, as formerly explained. Referring then to the tables of the elastic force of saturated vapour (given in a former lesson), at the temperature of the air under the bell-glass, the two terms of the ratio which represented the hygrometric state of the air, corresponding to the degree marked on the hygrometer, were determined, By repeating this experiment with solutions
more or less con: bulb 4 is about two-thirds full of ethet; in which is immersed centrated, and at the temperature of 109 Centigrade or 50°
a small thermometer enclosed in the tube. The two bulbs and Fahrenheit, he constructed the following table :
the tube are completely freed from air, by boiling the ether in
the bulb À, while the bulb B is still open, and then hermetically TABLE OF HYGROMETRIC STATES OF THE AIR, ANDICATED BY THE HAIR HYGRONETER, AT TIIE all the air: thus the tube and the bulb o contain only the
se ling the latter, when the vapour of the ether has driven out TEMPERATURE OF 50° FAHRENHEIT.
rapour of ether. The bulb is then covered with muslin, and Degrees of the Hygrometer.
Isgrometric states, ether is made to fall on it drop by drop. This liquid, by its 0°
evaporation, cools the hub and condenses the vapour in its 5
interior. The interior tension is then diminished, the ether 10
of the bulb a immediately gives out new vapour, which is 15
then condensed in the same manner in the other bulb; and 20
so on. Now in proportion as the liquid ilthus distilled fr m 25
the lower bulb into the upper bülb, the ether in the former is 30
when the air, which is in contact withi 35
the bulb A, and which is cooled with it, reaches the tempera40
ture at which the vapour of the water contained in it is suffi45
cient to saturate it, this vapour is condensed and deposited on 50
the bulb, A, in the form of dew, a ring of it surrounding the 55
surface of the liquid. It is there, in fact, that the cooling, 60
arising from the evaporation, is especially produced. The 65
ture of the dew-point, that is, the temperature of the surround: 0.500
ing air. In order to obtain this point to a greater degree of 75
approximation, we observe the temperature at the instant 80
when the deposited vapour disappears on being again Heated, 85
and we take the mean between this temperature and that at 90
which it was deposited. It is advisable; that during this 95
experiment; the hygrometer should be placed in a current of 100 1.000 air, as in an open window,
so that the
evaporation of the ether on the muslin may take place
with greater tapidity
. Lastly, This table shows that at 72°, the ait is only half saturated. in order to render the deposition of the dew more visible; As it is at this point that the index of the hygrometer generally the bulb A is commonly made of glass coloured black. As to stands at the surface of the ground, it is evident that, at the temperature
of the air, it is noted by means of a thermo: mean, the air contains the half of the vapout which it would meter placed on the stand of the apparatus. The hygrometer contain were it sáturated. In our climate the hygrometer 1 of Daniell having thus shown the temperature at which the
ait would be saturated, it is now required to deduce from it water. The tube e does not communicate with the exhauster; the hygrometric state of the air. For this purpose, we observe it contains only a thermometer intended to show the tempethat in a free space which contains a mixture of air and vapour rature of the air. Under this arrangement, some ether is at the atmospheric pressure, when the temperature is lowered, poured into the tube d, until it is about half-full, and then thë elastic force of the vapour remains constant until the point the stop-cock of the exhauster is opened; the water in the of saturation. In fact, the elastic force of the mixture is equal latter escapes, and the air in the tube D is rarefied. In conto the sum of the elastic forces of each fluid, as formerly sequence of the atmospheric pressure, the air then enters by explained ; now while the air is cooling, its tension remains the tube A; but as this air can only enter into the tube D and the same, for it increases as much by the diminution of its into the exhauster, by passing through the ether, it vaporises volume, as it decreases by the reduction of its temperature. à part of this liquid, and thus cools it sooner in proportion as
The tension of the vapour must, therefore, also remain the flow is more rapid. When the process of cooling reaches invariable, since the elastic force of the mixture nécessarily the point at which dew is deposited, as in the hygrometer of remains the same as that of the pressure of the atmosphere, Daniell, the thermometer T then indicates the corresponding after the process of cooling, as it was before. Consequently, temperature, and we have the elements necessary for deterwhen the air is cooled, the tension of the vapour which it con- mining the hygrometric state. In this instrument, the whole tains remains constant until the point of saturation; and at mass of ether is at the same temperature, in consequence of this point, this tension is the same as it was before the cooling the agitation of the current of air; besides, the observations commenced. According to this principle, if we look in the table are made at a distance by means of a telescope, and thus of elastic forces formerly given, p. the tension f corre- every cause of error is avoided. sponding to the dew-point, this tension will be precisely that Hygroscopes --Apparatus which merely indicate that there is possessed by the vapour of the water which is in the air at the more or less vapour of water in the air, without showing the moment of the experiment. If, therefore, we look in the same quantity; are called hygroscopes. They are constructed of table for the tension F of the vapour saturated, at the tempe- various sorts; the most in use are those made in the form of rature of the air, the quotient of the tension f divided by the little men whose heads are covered or uncovered with a hood, tension r will represent the hygrometric state of the air. For according as the air is more or less humid. These instruments example, the temperature of the air being 154 Centigrade, are constructed on the property which twisted cords and cat suppose that the thermometer of the bulb a marked 50 Centi- gut possess of untwisting by the action of humidity, and of grade at the moment whep the dew was deposited; then look twisting more by that of drought. Their indications are ing in the table of elastic forces p. 144 for tensions correspond- obtained by fixing to the figure a small piece of eat-gut by one ing to 5o and 150 Centigrade we find f equal to 0.267 inches of its extremities, and by attaching its other extremity to a and r equal to 0.500 inches, therefore = 0.614, is the moveable piece. These hygroscopes are slow in their motions;
that is, their indications are always behind the actual hygroratio of f to r, or the hygrometric state of the ait. The hygrometric state of the air, and their sensibility is very small. meter of Daniell is liable to several causes of error: 1st, the The Psychrometer. ---The psychrometer (cold measure, from the eraporation of the bulb A only cooling the liquid at the surface, Greek), inveäted by Professor August, enables us to ascertain the thermometer immersed in it cannot give exactly the tem" the tension of the vapour contained in the atmosphere, perature of the dew-point; 2nd, the observer standing near the dew-point; the point at which the atmosphere would be ke apparatus
, modifies the state of the surrounding air; ás saturated, and the absolute weight of the vapour contained in well as the temperature.
any particular volume of air. This instrument consists of two invented a condensation proids the causes oferrer in that de manien. Parthis apparatus with each other, and are divided in this or tenths or degrees, is composed of two thin polished silver cups of about two the scales ranging from about —25° Centigrade or -13• Fahinches in height and one inch in diameter, fig. 202.
renheit".0 40° Centigrade or 104° Fahrenheit. These thermoFig. 2022
meters are vertically fixed in á frame, at the distance of about three inches from each other. The bulb of the one is covered with muslin; which is kept continually moistened by means of
a cotton thread attached to it, the other end of the thread 7
being kept in a vessel or cúp full of distilled water; the bulb of the other is kept dry. As the water imbibed by the muslin surrounding the one bulb evaporates, the mercury in the thermometer begins to sink, and the drier that the air is, the more rapid will be the evaporation, and the more sensible the descent of the mercury. When the air around the bulb is saturated with moisture, the mercury will become stationary, and the point at which it rests will be the dew-point or condensation-point. The greater that the difference is between the heights
of the two thermometers, the more dry must be the state of the air, and the further is the vapour it contains from being at its maximum density. The difference between the heights of the two thermometers will be zero, if they are placed in an atmosphere containing aqueous vapou at its greatest đensity. From the degrees of heat shown by the two thermometers, the elastic force of the vapour in the air, and its amount per cent., can be determined.
The humidity or hygrometric state of the air varies at all hours, we might indeed say, at all minutes of the day. It is at its maximum before sun-rise, and at its minimum in ihe middle of the day. It varies in the different months of the year, In December, the air is the most humid; and in July and August, the most dry; yet, it can be proved that in these two latter
months the air contains the greatest quantity of the vapour of in these cups are fized two glass tubes d and . Each of stagnant water on the earth's surface being more active, non and
contains a very sensible thermometer listened by means sea-coasts, the air, other things being equal, is more humid that en The cork of the tube D, is traversed by a tube A in the interior of the continent. In the steppes, deserts, and e at both ends and immerscd in the cup to the bottom. pampas, the greatest known dryness prevails. On mountains, u ide stand and a leaden pipe, with an exhauster & full of I mean, it is more
humid there than in the plains. The relati
humidity varies with the wind; hence, we have dry winds and Prob. 5. Given 4x – 2y = 20, and 4x + 2y = 100; to find
Prob. 6. Given 52 +8=7y, and 5y + 32= 7*; to find
Here, let x be the greater; and y the less.
y = 4;
Therefore = 20. Ans. 20 and 4. (Continued from page 170.)
Prob. 8. To find one of two quantities, whose sum is equal
to h; and the difference of whose squares is equal to d. SIMPLE EQUATIONS.
Then x y=h TWO UNKNOWN QUANTITIES.
And 2 y =d
per question. In our former Lessons on Simple Equations, we gave the rules
From the first equation, we have, by transposition, for solving those which contain only one unknown quality; and,
x = h – y, with the exception of one or two, the whole Centenary of And by squaring both sides, we have, Problems were solved by means of these rules. We proceed now to show how to resolve equations which contain two un
q=h2 - 2y + y. known quantities.
From the second equation, we have, by transposition, Cases indeed frequently occur, in which two unknown quan
x = y +d; tities are necessarily introduced into the same calculation. Ex. Suppose the following equations are given, viz. :-
Now, by equating the two values of x”, we have
y? + d= 12 — 2hy + y*,
And by transposition and cancelling, we have
2hy = h — d; (1.) x = 14-Y
h - d
Whence, y = (2.) x=2+y.
2h Now, the first member of each of the equations is x, and
had 12 +0 the second member of each is equal to %. But according to
Therefore x =h
2h the Axiom that quantities which are respectively equal to another quantity, are equal to each other; therefore we have Prob. 9. Given ax + by=h, and xty=d; to find the
values of x and y. 2+y=14 — y; whence, y = 6. Lastly, by substituting the value of y in the 1st equation,
Here, from the first equation, we have, by transposition, we have a +6= 14; and x = 8. Therefore, 8 and 6 are the
ax = h – by, values of x and y.
h - by
And In solving the preceding problem, it will be observed that we first found the value of the unknown quantity x, in each equation ; and then, by making one of the expressions denot
Again, from the second equation, we have, by transposition, ing the value of x, equal to the other, we formed a new equa
x=dy, tion, which contained only the other unknown quantity y.
እ h - by This process is called extermination or elimination,
=d-y, In the resolution of equations, there are three methods of
h - by = ad - ay,
And, ay by = ad - h;
From this equation, by separating the left hand member into comparison.
(a - b) y=ad - h; RULE.—Find the value of one of the unknown quantities in each
adh of the equations, and form a new equation by making one of these
Whence, values equal to the other. Find the value of the unknown quantity in this equation, by the rules formerly given. Then substitute this
ad h h - bd value of the one unknown quantity either of the other equations, Consequently, s = d. and resolving it by the same rules, the other unknown quantity will be found.
The rule given above may be generally applied for the ex: Prob. 1. Given x+y= 36, and x - y=12; to find the termination of unknown quantities. But there are cases in values of x and y.
which other methods will be found more expeditious.
x=hy, and ax + 6*°= y*; to find the Transposing y in the first equation, gives * = 36
values of x and y. Transposing y in the second equation, , * = 12 + y As in the first of these equations, a is equal to hy, we may Making these values of x equal
12 +y=36 –
in the second equation substitute this value of x for 2 itself, Transposing, etc.,
The second equation will then become, any + bhy Ey Substituting the value of y,
x = 12 + 12 = 24.
The equality of the two sides is not affected by this alteraHence, 24 and 12 are the values required.
tion, because we only change one quantity & for another which Prob. 2. Given 2x + 3y=28, and 3x +2y=27; to find is equal to it. By this means we obtain an equation which the values of x and y. Ans, 5 and 6.
contains only one unknown
quantity. Whence, y = ah+bh,
This process is called extermination by substitution,
y = 12
y = 4.
RULE.— Find the value of one of the unknown quantities, in one are always to be increased or diminished alike, in order to
Prob. 22. Given 2x + 4y = 20, and 4x + 5y = 28; to find
Here, multiplying the first equation by 2, we have
4x + 8y = 40;
Subtracting the second equation from this, we have
3y = 12,
Whence y = 4; and I = 2.
Prob. 23. Given 2x +y=16, and 3x
- 3y = 6; to find
the values of x and y. Ans. 6 and 4. Whence, ny transposition, etc.,
Prob. 24. Given 4x + 3y = 50, and 3x – 3y = 6; to find
the values of x and y. Ans. 8 and 6. And, from the first equation,
Prob. 25. Given 3x +y = 38, and 5x + 4y =68; to find
the values of x and y, Ans. 12 and 2.
- 4y, and 6x - 63 = - 77:
Prob. 27. The numbers of two opposing armies are such,
Prob. 14. Given 3x + 3y=72, and 4x + 5y = 116; to find greater army, added to three times the number in the less,
is 52219. What is the number in each army? Ans. 11111
Prob, 28. The sum of two numbers is 220, and if 3 times the
less be taken from 4 times the greater, the remainder will be
three rules for exterminating unknown quantities may be used
The student will find it a useful exercise to solve each ex.
ample by each of the several methods, and carefully to observe
Prob. 29. The mast of a ship consists of two parts; one-
by six times the upper part, is equal to 12 feet. What is the
height of the mast? Ans, 108 feet.
Prob. 30. To find a fraction such that, if a unit be added to
the numerator, the fraction will be equal to }; but if a unit
Let x = the numerator, and y=the denominator, 2x = a + b,
Here, by the first condition, we have
And by the second,
x = 4, the numerato , Here, if we subtract the second equation from the first, we
y = 15, the denominator. shall have =h-d, where y is exterminated, without affect
Ans. ing the equality of the sides. Whence, y = 3d - 2h.
Prob. 31. What two numbers are those, whose difference is - 2y = a, and x + 4y = b; to find the to their sum as 2 to 3; and whose sum is to their product as 3
to 5: Ans. 10 and 2. Here, multiplying the first equation by 2, we have
Prob. 32. To find two numbers such, that the product of 2x - 4y = 2a,
their bum and difference shall be 5, and the product of the Then, adding the second and third equations, we have
sum of their squares and the difference of their squares shall
be 65. Ans. 3 and 2. 3x = 6 + 2a,
Prob. 33. To find two numbers whose sum is 32, and whose =$(6+ 2a)
product is 240. Ans. 20 and 12.
Prob. 34. To find two numbers whose sum is 52, and the
Prob. 35. A certain number consists of two digits or figures,
the sum of which is 8. If 36 be added to the number, the
Prob. 36, The united ages of A and B amount to a certain
Prob. 37. A merchant having mixed a quantity of brandy
compound would have contained 7 gallons of brandy for every It must be kept in mind that both members of an equation 6 of gin; but if he had put in 6 gallons less of each, the pro.
Talues of x and y.
values of u and y.
Prob. 21. Given t values of x and y.
Tesolved as before.
portions would have been as 6 to 5. How many gallons did To find x and y we have only to take their values from the he mix of each? Ans. 78 and 66.
third and fifth equations. Reducing the fifth, we have
=9–259 - 5 = 4; THREE UNKNOWN QUANTITIES.
Transposing in the third, we have In the preceding examples of two unknown quantities, it
X = 12 will be perceived that the conditions of each problem have
2-y= 12–5–4= 3. furnished two equations independent of each other. It often
Prob 40. Given x+y+z=12, x + 2y + 3 = 20, and becomes necessary to introduce three or more unknown quanti- 3x + dy +:=6; to tind the values of , ý and .. Ans. 6, 4 ties into a calculation. In such cases, if the problem admits and 2. of a determinate answer, there will always arise from the con- In many of the examples in the preceding lessons, the proditions as many equations independent of each other, as there cesses might have been shortened. But the object was to are unknown quantities. Equations are said to be independent, when they express of expeditious solutions. The learner will do well, as he passes
illustrate general principles, rather than to furnish specimens different conditions,
along, to exercise his skill in abridging the calculations here They are said to be dependent when they express the same given, or substituting others in their stead. conditions under different forms, The former are not convertible into each other; but the latter may be changed from Prob. 41. Given x+y=a, a +?=b, and y+=c; to one form into the other. Thus 1 x=y; and b=1+ x, are find the values of x, y and e. Ans. x=} (a +6-e); y= dependent equations, because one is formed from the other by 1 (a −6+c), and ==})- a +6+c). merely transposing x. Equations are said to be identical when they express the same thing in the same form expressed or 100 dollars, but neither is able to pay for the whole. The
Prob. 42. Three persons, A, B and C, purchase a horse for implied; as 4x − 6 = 4x – 6, or 2(+ — 3) = 4x – 6.
payment would require the whole of A's money, together Prob. 38. Given x+y+z=12, x + 2y — 2= 10, and with half of B's; or the whole of B's with one third of C's; 2 ty-z= 4; to find the values of x, y and z.
or the whole of C's, with one fourth of A's. How much
money had each ? Ans. 64, 72 and 84 dollars, From these three equations, two others may be derived which shall contain only two unknown quantities. One of the
The learner must exercise his own judgment as to the three unknown quantities in the original equations may be choice of the quantity to be first exterminated. It will geneexterminated, in the same manner as when there are at first rally be best to begin with that which is most free from co-effionly two, by the rules already given. Thus, if in the equations cients, fractions, radical signs, etc. that is, the least involved. given above, we transpose y and e, we shall have,
Prob. 43. The sum of the distances which three persons, A, From the first, = 12 - Y
B, and C have travelled, is 62 miles ; A's distance is equal to the second, x = 10 - 2y + 2%.
4 times C's added to twice B's; and twice A's added to 3 the third, = 4-ytu.
times B’s, is equal to 17 times C's. What are the respective
distances ? Ans. 46, 9 and 7. From these we may now deduce two new equations, from which a shall be excluded.
Prob. 44. Given jx + 3y + 12 = 62, fx + y + x= 47, By making the first and second equal, we have
and 1x + y + =38; to find the values of x, y and z. Ans.
24, 60, and 120. 12 -- Y - Z=10 – 2y + 23.
Prob. 45. Given xy = 600, xz = 300, and yz = 200; to find By making the second and third equal, we have
the values of x, y and . Ans. 30, 20 and 10. 10 – 2y + 22 = 4–4 + z. Reducing the first of these two, we have
y = 32 — 2. Reducing the second, we have
LESSONS IN READING AND ELOCUTION. y=.+ 6.
No. XV. From these two equations one may be derived containing only one unknown quantity.
EXERCISES ON EXPRESSIVE TONE (continued). By making the one equal to the other, we have
I[l.-THE PURITANS. 32 – 2 =% + 6 Therefore, : = 4. Hence, y = 10, and x = -2.
[Marked for Inflections.) To solve a problem containing three unknown quantities, | The Puritans were inen whose minds had derived a peculiar and producing three independent equations,
character from the daily contemplation of superior beings and Rule_First, from the three equations deduce two, containing
etérnal interests. Not content with acknowledging, in general only two unknnin quantities . Then, from these two deduce one; érery' event
to the will of the Great
Being, for whose power
terms, an overruling Providence, they habilually ascribed containing only one unknoun quantity. Lastly, find the ralucs of nothing was too vást, for whose inspéction nothing was too the other unknown quantities as before.
minute. To know Hlin, to serve Him, to enjoy Him, yas with For making these reductions, the rules already given are them, the great end of existence. They rejected with contempt, sufficient,
the ceremonious homage which other sects substituted for the Prob. 39. Given x + 5y + 6= 53, x + 3y +37= 30, and pure worship of the soul. Instead of catching occasional . a+y+= 12; to find the values of x, y and .
glimpses of the Deity through an obscìring veil, they aspired
to gaze full on the iniólerable brightness, and to commune with Here, from these three equations in order to deriye tvo Him, fáce to face. Ilence originated their contempt for terrescontaining only two unknown quantities,
trial distinctions. The difference between the grèatest and Subtracting the second from the first, we have
méanest of mankind seemed to vanish, when compared with 2y + 3z = 23.
the boundless interval which separated the whole råce from
Him on whom their own eyes were constantly fixed. They Subtracting the third from the second, we have
recognised no title to superiority but His fà vour, and confi2y + 2z = 18.
dent of that favour, they despised all the accomplishments and Next, from these two, in order to derive one,
all the dignities of the world. If they were unacquainted
with the works of philòsophers and poets, they were deeply Subtracting the fifth from the fourth, we have
réad in the oracles of God. If their names were not found in 5.
the règisters of héralds, they felt assured that they were